Math Functions - Java Programming

When building applications that need calculations like distance between points, interest rates, or game physics, you need reliable mathematical operations. Java's Math class provides static methods for common mathematical computations including powers, roots, trigonometry, and rounding.

Math class A utility class providing static methods for mathematical operations like powers, roots, and trigonometry.

Basic Operations

The Math class provides fundamental mathematical operations.

                            Basic.java
                        
Replay: real traced execution (multi-file project)

                            // Basic math operations

public class Basic {
    public static void main(String[] args) {
        // Absolute value
        System.out.println("Absolute value:");
        System.out.println("abs(-5): " + Math.abs(-5));
        System.out.println("abs(-3.14): " + Math.abs(-3.14));
        System.out.println("abs(7): " + Math.abs(7));

        // Power
        System.out.println("\nPower:");
        System.out.println("pow(2, 3): " + Math.pow(2, 3));
        System.out.println("pow(5, 2): " + Math.pow(5, 2));
        System.out.println("pow(2, 10): " + Math.pow(2, 10));
        System.out.println("pow(3, 0): " + Math.pow(3, 0));

        // Square root
        System.out.println("\nSquare root:");
        System.out.println("sqrt(16): " + Math.sqrt(16));
        System.out.println("sqrt(2): " + Math.sqrt(2));
        System.out.println("sqrt(100): " + Math.sqrt(100));
        System.out.println("sqrt(0.25): " + Math.sqrt(0.25));

        // Cube root
        System.out.println("\nCube root:");
        System.out.println("cbrt(8): " + Math.cbrt(8));
        System.out.println("cbrt(27): " + Math.cbrt(27));
        System.out.println("cbrt(64): " + Math.cbrt(64));

        // Max and min
        System.out.println("\nMax and min:");
        System.out.println("max(5, 10): " + Math.max(5, 10));
        System.out.println("min(5, 10): " + Math.min(5, 10));
        System.out.println("max(-3, -7): " + Math.max(-3, -7));
        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));

        // Sign
        System.out.println("\nSign (signum):");
        System.out.println("signum(5): " + Math.signum(5));     // 1.0
        System.out.println("signum(-5): " + Math.signum(-5));   // -1.0
        System.out.println("signum(0): " + Math.signum(0));     // 0.0

        // Exponential and logarithm
        System.out.println("\nExponential and logarithm:");
        System.out.println("exp(1): " + Math.exp(1));           // e^1
        System.out.println("exp(2): " + Math.exp(2));           // e^2
        System.out.println("log(Math.E): " + Math.log(Math.E)); // ln(e) = 1
        System.out.println("log10(100): " + Math.log10(100));   // log base 10

        // Constants
        System.out.println("\nMath constants:");
        System.out.println("PI: " + Math.PI);
        System.out.println("E: " + Math.E);

        // Practical examples
        System.out.println("\nPractical examples:");

        // Distance
        double x1 = 0, y1 = 0, x2 = 3, y2 = 4;
        double distance = Math.sqrt(Math.pow(x2 - x1, 2) + Math.pow(y2 - y1, 2));
        System.out.println("Distance from (0,0) to (3,4): " + distance);

        // Circle area
        double radius = 5;
        double area = Math.PI * Math.pow(radius, 2);
        System.out.println("Circle area (r=5): " + area);

        // Compound interest
        double principal = 1000;
        double rate = 0.05;
        int years = 10;
        double amount = principal * Math.pow(1 + rate, years);
        System.out.println("Compound interest: $" + amount);
    }

    //help h1
    // Math.abs(x) - absolute value
    // Math.pow(base, exp) - power
    // Math.sqrt(x) - square root
    // Math.cbrt(x) - cube root
    // Math.max(a, b), Math.min(a, b)
    // Math.exp(x) - e^x
    // Math.log(x) - natural log
    // Math.PI, Math.E - constants
    //end
}

                        
1y2 ← 4.0, distance ← 5.0, radius ← 5.0, area ← 78.53981633974483⋯3public class Basic {4    public static void main(String[] args) {5        // Absolute value6        System.out.println("Absolute value:");7        System.out.println("abs(-5): " + Math.abs(-5));8        System.out.println("abs(-3.14): " + Math.abs(-3.14));9        System.out.println("abs(7): " + Math.abs(7));1011        // Power12        System.out.println("\nPower:");13        System.out.println("pow(2, 3): " + Math.pow(2, 3));14        System.out.println("pow(5, 2): " + Math.pow(5, 2));15        System.out.println("pow(2, 10): " + Math.pow(2, 10));16        System.out.println("pow(3, 0): " + Math.pow(3, 0));1718        // Square root19        System.out.println("\nSquare root:");20        System.out.println("sqrt(16): " + Math.sqrt(16));21        System.out.println("sqrt(2): " + Math.sqrt(2));22        System.out.println("sqrt(100): " + Math.sqrt(100));23        System.out.println("sqrt(0.25): " + Math.sqrt(0.25));2425        // Cube root26        System.out.println("\nCube root:");27        System.out.println("cbrt(8): " + Math.cbrt(8));28        System.out.println("cbrt(27): " + Math.cbrt(27));29        System.out.println("cbrt(64): " + Math.cbrt(64));3031        // Max and min32        System.out.println("\nMax and min:");33        System.out.println("max(5, 10): " + Math.max(5, 10));34        System.out.println("min(5, 10): " + Math.min(5, 10));35        System.out.println("max(-3, -7): " + Math.max(-3, -7));36        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));3738        // Sign39        System.out.println("\nSign (signum):");40        System.out.println("signum(5): " + Math.signum(5));     // 1.041        System.out.println("signum(-5): " + Math.signum(-5));   // -1.042        System.out.println("signum(0): " + Math.signum(0));     // 0.04344        // Exponential and logarithm45        System.out.println("\nExponential and logarithm:");46        System.out.println("exp(1): " + Math.exp(1));           // e^147        System.out.println("exp(2): " + Math.exp(2));           // e^248        System.out.println("log(Math.E): " + Math.log(Math.E)); // ln(e) = 149        System.out.println("log10(100): " + Math.log10(100));   // log base 105051        // Constants52        System.out.println("\nMath constants:");53        System.out.println("PI: " + Math.PI);54        System.out.println("E: " + Math.E);5556        // Practical examples57        System.out.println("\nPractical examples:");58        59        // Distance60        double x1 = 0, y1 = 0, x2 = 3, y2→ 4.0 = 4;61        double distance→ 5.0 = Math.sqrt(Math.pow(x23.0 - x10.0, 2) + Math.pow(y24.0 - y10.0, 2));62        System.out.println("Distance from (0,0) to (3,4): " + distance5.0);6364        // Circle area65        double radius→ 5.0 = 5;66        double area→ 78.53981633974483 = Math.PI * Math.pow(radius5.0, 2);67        System.out.println("Circle area (r=5): " + area78.53981633974483);6869        // Compound interest70        double principal→ 1000.0 = 1000;71        double rate→ 0.05 = 0.05;72        int years→ 10 = 10;73        double amount→ 1628.894626777442 = principal1000.0 * Math.pow(1 + rate0.05, years10);74        System.out.println("Compound interest: $" + amount1628.894626777442);75    }⋯outputAbsolute value:
abs(-5): 5
abs(-3.14): 3.14
abs(7): 7

Power:
pow(2, 3): 8.0
pow(5, 2): 25.0
pow(2, 10): 1024.0
pow(3, 0): 1.0

Square root:
sqrt(16): 4.0
sqrt(2): 1.4142135623730951
sqrt(100): 10.0
sqrt(0.25): 0.5

Cube root:
cbrt(8): 2.0
cbrt(27): 3.0
cbrt(64): 4.0

Max and min:
max(5, 10): 10
min(5, 10): 5
max(-3, -7): -3
min(3.14, 2.71): 2.71

Sign (signum):
signum(5): 1.0
signum(-5): -1.0
signum(0): 0.0

Exponential and logarithm:
exp(1): 2.718281828459045
exp(2): 7.38905609893065
log(Math.E): 1.0
log10(100): 2.0

Math constants:
PI: 3.141592653589793
E: 2.718281828459045

Practical examples:
Distance from (0,0) to (3,4): 5.0
Circle area (r=5): 78.53981633974483
Compound interest: $1628.894626777442

Static methods Methods called on the class itself (Math.pow()) rather than on an instance, since Math has no state to maintain.

Rounding Functions

Control how decimal numbers convert to integers.

                            Rounding.java
                        
Replay: real traced execution (multi-file project)

                            // Rounding operations

public class Rounding {
    public static void main(String[] args) {
        double value = 3.7;

        System.out.println("Value: " + value);
        System.out.println();

        // Round
        System.out.println("round():");
        System.out.println("round(3.7): " + Math.round(3.7));   // 4
        System.out.println("round(3.4): " + Math.round(3.4));   // 3
        System.out.println("round(3.5): " + Math.round(3.5));   // 4
        System.out.println("round(-3.5): " + Math.round(-3.5)); // -3
        System.out.println("round(-3.6): " + Math.round(-3.6)); // -4

        // Ceil (round up)
        System.out.println("\nceil() - round up:");
        System.out.println("ceil(3.1): " + Math.ceil(3.1));     // 4.0
        System.out.println("ceil(3.9): " + Math.ceil(3.9));     // 4.0
        System.out.println("ceil(-3.1): " + Math.ceil(-3.1));   // -3.0
        System.out.println("ceil(5.0): " + Math.ceil(5.0));     // 5.0

        // Floor (round down)
        System.out.println("\nfloor() - round down:");
        System.out.println("floor(3.1): " + Math.floor(3.1));   // 3.0
        System.out.println("floor(3.9): " + Math.floor(3.9));   // 3.0
        System.out.println("floor(-3.1): " + Math.floor(-3.1)); // -4.0
        System.out.println("floor(5.0): " + Math.floor(5.0));   // 5.0

        // Compare all three
        System.out.println("\nCompare rounding methods:");
        double[] testValues = {2.3, 2.5, 2.7, -2.3, -2.5, -2.7};

        System.out.println("Value\tFloor\tRound\tCeil");
        for (double v : testValues) {
            System.out.printf("%.1f\t%.1f\t%d\t%.1f%n",
                v, Math.floor(v), Math.round(v), Math.ceil(v));
        }

        // Rounding to decimal places
        System.out.println("\nRound to decimal places:");
        double pi = Math.PI;
        System.out.println("Original: " + pi);
        System.out.println("2 decimals: " + roundToDecimals(pi, 2));
        System.out.println("4 decimals: " + roundToDecimals(pi, 4));

        // Truncate (toward zero)
        System.out.println("\nTruncate (cast to int):");
        System.out.println("(int) 3.9: " + (int) 3.9);     // 3
        System.out.println("(int) -3.9: " + (int) -3.9);   // -3

        // Practical examples
        System.out.println("\nPractical examples:");

        // Calculate pages needed
        int items = 47;
        int itemsPerPage = 10;
        int pages = (int) Math.ceil((double) items / itemsPerPage);
        System.out.println(items + " items, " + itemsPerPage + " per page = " + pages + " pages");

        // Round money
        double price = 19.996;
        double rounded = roundToDecimals(price, 2);
        System.out.println("Price $" + price + " rounded: $" + rounded);

        // Nearest multiple
        int number = 47;
        int multiple = 10;
        int nearest = (int) (Math.round((double) number / multiple) * multiple);
        System.out.println(number + " to nearest " + multiple + ": " + nearest);

        // Division with rounding
        System.out.println("\nDivision with rounding:");
        System.out.println("7 / 2 (floor): " + 7 / 2);
        System.out.println("7 / 2.0 (round): " + Math.round(7 / 2.0));
        System.out.println("7 / 2.0 (ceil): " + (int) Math.ceil(7 / 2.0));
    }
    public static double roundToDecimals(double value, int decimals) {
        double scale = Math.pow(10, decimals);
        return Math.round(value * scale) / scale;
    }

    //help h1
    // Math.round(x) - round to nearest integer
    // Math.ceil(x) - round up to integer
    // Math.floor(x) - round down to integer
    // (int) x - truncate (toward zero)
    // Round to N decimals: round(x * 10^N) / 10^N
    // ceil() useful for pagination
    //end
}

                        
1value ← 3.7⋯3public class Rounding {4    public static void main(String[] args) {5        double value→ 3.7 = 3.7;6        7        System.out.println("Value: " + value3.7);8        System.out.println();910        // Round11        System.out.println("round():");12        System.out.println("round(3.7): " + Math.round(3.7));   // 413        System.out.println("round(3.4): " + Math.round(3.4));   // 314        System.out.println("round(3.5): " + Math.round(3.5));   // 415        System.out.println("round(-3.5): " + Math.round(-3.5)); // -316        System.out.println("round(-3.6): " + Math.round(-3.6)); // -41718        // Ceil (round up)19        System.out.println("\nceil() - round up:");20        System.out.println("ceil(3.1): " + Math.ceil(3.1));     // 4.021        System.out.println("ceil(3.9): " + Math.ceil(3.9));     // 4.022        System.out.println("ceil(-3.1): " + Math.ceil(-3.1));   // -3.023        System.out.println("ceil(5.0): " + Math.ceil(5.0));     // 5.02425        // Floor (round down)26        System.out.println("\nfloor() - round down:");27        System.out.println("floor(3.1): " + Math.floor(3.1));   // 3.028        System.out.println("floor(3.9): " + Math.floor(3.9));   // 3.029        System.out.println("floor(-3.1): " + Math.floor(-3.1)); // -4.030        System.out.println("floor(5.0): " + Math.floor(5.0));   // 5.03132        // Compare all three33        System.out.println("\nCompare rounding methods:");34        double[] testValues = {2.3, 2.5, 2.7, -2.3, -2.5, -2.7};35        36        System.out.println("Value\tFloor\tRound\tCeil");37        for (double v : testValues) {⋯outputValue: 3.7
round():
round(3.7): 4
round(3.4): 3
round(3.5): 4
round(-3.5): -3
round(-3.6): -4

ceil() - round up:
ceil(3.1): 4.0
ceil(3.9): 4.0
ceil(-3.1): -3.0
ceil(5.0): 5.0

floor() - round down:
floor(3.1): 3.0
floor(3.9): 3.0
floor(-3.1): -4.0
floor(5.0): 5.0

Compare rounding methods:
Value	Floor	Round	Ceil
2for (double v : testValues)pass 1 of 6⋯36System.out.println("Value\tFloor\tRound\tCeil");37for (double v2.3 : testValues) {38    System.out.printf("%.1f\t%.1f\t%d\t%.1f%n",39        v2.3, Math.floor(v), Math.round(v), Math.ceil(v));40}⋯All 6 passes — pass 1 is the card abovepassv
12.3
22.5
32.7
4-2.3
5-2.5
6-2.7
3pi ← 3.141592653589793⋯42// Rounding to decimal places43System.out.println("\nRound to decimal places:");44double pi→ 3.141592653589793 = Math.PI;45System.out.println("Original: " + pi3.141592653589793);46System.out.println("2 decimals: " + roundToDecimals(pi3.141592653589793, 2));47System.out.println("4 decimals: " + roundToDecimals(pi, 4));⋯output
Round to decimal places:
Original: 3.141592653589793
4scale ← 100.0pass 1 of 3⋯79}80public static double roundToDecimals(double value3.141592653589793, int decimals2) {81    double scale→ 100.0 = Math.pow(10, decimals2);82    return Math.round(value3.141592653589793 * scale100.0) / scale;83}⋯All 3 passes — pass 1 is the card abovepassvaluedecimalsscale
13.1415926535897932100.0
23.141592653589793410000.0
319.9962100.0
5System.out.println("2 decimals: " + roundToDecimals(pi, 2));⋯45System.out.println("Original: " + pi);46System.out.println("2 decimals: " + roundToDecimals(pi3.141592653589793, 2));47System.out.println("4 decimals: " + roundToDecimals(pi3.141592653589793, 4));⋯output2 decimals: 3.14
6items ← 47, itemsPerPage ← 10, pages ← 5, price ← 19.996⋯46System.out.println("2 decimals: " + roundToDecimals(pi, 2));47System.out.println("4 decimals: " + roundToDecimals(pi3.141592653589793, 4));4849// Truncate (toward zero)50System.out.println("\nTruncate (cast to int):");51System.out.println("(int) 3.9: " + (int) 3.9);     // 352System.out.println("(int) -3.9: " + (int) -3.9);   // -35354// Practical examples55System.out.println("\nPractical examples:");5657// Calculate pages needed58int items→ 47 = 47;59int itemsPerPage→ 10 = 10;60int pages→ 5 = (int) Math.ceil((double) items / itemsPerPage10);61System.out.println(items47 + " items, " + itemsPerPage10 + " per page = " + pages5 + " pages");6263// Round money64double price→ 19.996 = 19.996;65double rounded = roundToDecimals(price19.996, 2);66System.out.println("Price $" + price + " rounded: $" + rounded);⋯output4 decimals: 3.1416

Truncate (cast to int):
(int) 3.9: 3
(int) -3.9: -3

Practical examples:
47 items, 10 per page = 5 pages
7rounded ← 20.0, number ← 47, multiple ← 10, nearest ← 50⋯64    double price = 19.996;65    double rounded→ 20.0 = roundToDecimals(price19.996, 2);66    System.out.println("Price $" + price19.996 + " rounded: $" + rounded20.0);6768    // Nearest multiple69    int number→ 47 = 47;70    int multiple→ 10 = 10;71    int nearest→ 50 = (int) (Math.round((double) number / multiple10) * multiple);72    System.out.println(number47 + " to nearest " + multiple10 + ": " + nearest50);7374    // Division with rounding75    System.out.println("\nDivision with rounding:");76    System.out.println("7 / 2 (floor): " + 7 / 2);77    System.out.println("7 / 2.0 (round): " + Math.round(7 / 2.0));78    System.out.println("7 / 2.0 (ceil): " + (int) Math.ceil(7 / 2.0));79}⋯outputPrice $19.996 rounded: $20.0
47 to nearest 10: 50

Division with rounding:
7 / 2 (floor): 3
7 / 2.0 (round): 4
7 / 2.0 (ceil): 4

All 6 passes — pass 1 is the card above
pass	`v`
1	2.3
2	2.5
3	2.7
4	-2.3
5	-2.5
6	-2.7

All 3 passes — pass 1 is the card above
pass	`value`	`decimals`	`scale`
1	3.141592653589793	2	100.0
2	3.141592653589793	4	10000.0
3	19.996	2	100.0

Comparison Functions

Find minimum and maximum values efficiently.

                            score
                            score = 150 (default)score = -10score = 50
                        

                            Comparison.java
                        
Replay: real traced execution (multi-file project)

                            // Comparison and clamping

public class Comparison {
    public static void main(String[] args) {
        // Max and min
        System.out.println("Max and min:");
        System.out.println("max(5, 10): " + Math.max(5, 10));
        System.out.println("min(5, 10): " + Math.min(5, 10));
        System.out.println("max(-5, -10): " + Math.max(-5, -10));
        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));

        // Max/min of multiple values
        System.out.println("\nMax/min of multiple:");
        int[] numbers = {5, 2, 9, 1, 7, 3};
        int maxValue = findMax(numbers);
        int minValue = findMin(numbers);
        System.out.println("Array: " + java.util.Arrays.toString(numbers));
        System.out.println("Max: " + maxValue);
        System.out.println("Min: " + minValue);

        // Clamp (restrict to range)
        System.out.println("\nClamp to range [0, 100]:");
        System.out.println("clamp(-10): " + clamp(-10, 0, 100));
        System.out.println("clamp(50): " + clamp(50, 0, 100));
        System.out.println("clamp(150): " + clamp(150, 0, 100));

        // Sign comparison
        System.out.println("\nSign comparison:");
        System.out.println("signum(5): " + Math.signum(5));
        System.out.println("signum(-5): " + Math.signum(-5));
        System.out.println("signum(0): " + Math.signum(0));

        // Same sign check
        System.out.println("\nSame sign:");
        System.out.println("5 and 10: " + sameSign(5, 10));
        System.out.println("5 and -10: " + sameSign(5, -10));
        System.out.println("-5 and -10: " + sameSign(-5, -10));

        // Absolute difference
        System.out.println("\nAbsolute difference:");
        System.out.println("|5 - 10|: " + Math.abs(5 - 10));
        System.out.println("|10 - 5|: " + Math.abs(10 - 5));
        System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));

        // Within tolerance
        System.out.println("\nWithin tolerance:");
        double a = 3.14159;
        double b = 3.14;
        double tolerance = 0.01;
        System.out.println(a + " ≈ " + b + " (±" + tolerance + "): " +
                          withinTolerance(a, b, tolerance));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Keep score in bounds
        int score = 150;
        int bounded = clamp(score, 0, 100);
        System.out.println("Score " + score + " bounded to [0,100]: " + bounded);

        // Volume control
        double volume = 1.5;
        double validVolume = clamp(volume, 0.0, 1.0);
        System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);

        // Temperature range
        int temp = -5;
        int minTemp = 0;
        int maxTemp = 30;
        int validTemp = clamp(temp, minTemp, maxTemp);
        System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);

        // Find range
        int[] values = {23, 45, 12, 67, 34};
        int min = findMin(values);
        int max = findMax(values);
        int range = max - min;
        System.out.println("\nValues: " + java.util.Arrays.toString(values));
        System.out.println("Range: " + min + " to " + max + " (span: " + range + ")");

        // Midpoint
        System.out.println("\nMidpoint:");
        System.out.println("Between 10 and 20: " + midpoint(10, 20));
        System.out.println("Between -5 and 15: " + midpoint(-5, 15));
    }
    public static int findMax(int[] arr) {
        int max = arr[0];
        for (int val : arr) {
            max = Math.max(max, val);
        }
        return max;
    }
    public static int findMin(int[] arr) {
        int min = arr[0];
        for (int val : arr) {
            min = Math.min(min, val);
        }
        return min;
    }
    public static int clamp(int value, int min, int max) {
        return Math.max(min, Math.min(max, value));
    }

    public static double clamp(double value, double min, double max) {
        return Math.max(min, Math.min(max, value));
    }
    public static boolean sameSign(double a, double b) {
        return Math.signum(a) == Math.signum(b);
    }
    public static boolean withinTolerance(double a, double b, double tolerance) {
        return Math.abs(a - b) <= tolerance;
    }
    public static double midpoint(double a, double b) {
        return (a + b) / 2;
    }

    //help h1
    // Math.max(a, b), Math.min(a, b)
    // Clamp: max(min, min(max, value))
    // Math.signum(x) - sign of number
    // Math.abs(a - b) - absolute difference
    // Within tolerance: abs(a - b) <= tol
    // Iterate for max/min of array
    //end
}
// Comparison and clamping

public class Comparison {
    public static void main(String[] args) {
        // Max and min
        System.out.println("Max and min:");
        System.out.println("max(5, 10): " + Math.max(5, 10));
        System.out.println("min(5, 10): " + Math.min(5, 10));
        System.out.println("max(-5, -10): " + Math.max(-5, -10));
        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));

        // Max/min of multiple values
        System.out.println("\nMax/min of multiple:");
        int[] numbers = {5, 2, 9, 1, 7, 3};
        int maxValue = findMax(numbers);
        int minValue = findMin(numbers);
        System.out.println("Array: " + java.util.Arrays.toString(numbers));
        System.out.println("Max: " + maxValue);
        System.out.println("Min: " + minValue);

        // Clamp (restrict to range)
        System.out.println("\nClamp to range [0, 100]:");
        System.out.println("clamp(-10): " + clamp(-10, 0, 100));
        System.out.println("clamp(50): " + clamp(50, 0, 100));
        System.out.println("clamp(150): " + clamp(150, 0, 100));

        // Sign comparison
        System.out.println("\nSign comparison:");
        System.out.println("signum(5): " + Math.signum(5));
        System.out.println("signum(-5): " + Math.signum(-5));
        System.out.println("signum(0): " + Math.signum(0));

        // Same sign check
        System.out.println("\nSame sign:");
        System.out.println("5 and 10: " + sameSign(5, 10));
        System.out.println("5 and -10: " + sameSign(5, -10));
        System.out.println("-5 and -10: " + sameSign(-5, -10));

        // Absolute difference
        System.out.println("\nAbsolute difference:");
        System.out.println("|5 - 10|: " + Math.abs(5 - 10));
        System.out.println("|10 - 5|: " + Math.abs(10 - 5));
        System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));

        // Within tolerance
        System.out.println("\nWithin tolerance:");
        double a = 3.14159;
        double b = 3.14;
        double tolerance = 0.01;
        System.out.println(a + " ≈ " + b + " (±" + tolerance + "): " +
                          withinTolerance(a, b, tolerance));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Keep score in bounds
        int score = -10;
        int bounded = clamp(score, 0, 100);
        System.out.println("Score " + score + " bounded to [0,100]: " + bounded);

        // Volume control
        double volume = 1.5;
        double validVolume = clamp(volume, 0.0, 1.0);
        System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);

        // Temperature range
        int temp = -5;
        int minTemp = 0;
        int maxTemp = 30;
        int validTemp = clamp(temp, minTemp, maxTemp);
        System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);

        // Find range
        int[] values = {23, 45, 12, 67, 34};
        int min = findMin(values);
        int max = findMax(values);
        int range = max - min;
        System.out.println("\nValues: " + java.util.Arrays.toString(values));
        System.out.println("Range: " + min + " to " + max + " (span: " + range + ")");

        // Midpoint
        System.out.println("\nMidpoint:");
        System.out.println("Between 10 and 20: " + midpoint(10, 20));
        System.out.println("Between -5 and 15: " + midpoint(-5, 15));
    }
    public static int findMax(int[] arr) {
        int max = arr[0];
        for (int val : arr) {
            max = Math.max(max, val);
        }
        return max;
    }
    public static int findMin(int[] arr) {
        int min = arr[0];
        for (int val : arr) {
            min = Math.min(min, val);
        }
        return min;
    }
    public static int clamp(int value, int min, int max) {
        return Math.max(min, Math.min(max, value));
    }

    public static double clamp(double value, double min, double max) {
        return Math.max(min, Math.min(max, value));
    }
    public static boolean sameSign(double a, double b) {
        return Math.signum(a) == Math.signum(b);
    }
    public static boolean withinTolerance(double a, double b, double tolerance) {
        return Math.abs(a - b) <= tolerance;
    }
    public static double midpoint(double a, double b) {
        return (a + b) / 2;
    }

    //help h1
    // Math.max(a, b), Math.min(a, b)
    // Clamp: max(min, min(max, value))
    // Math.signum(x) - sign of number
    // Math.abs(a - b) - absolute difference
    // Within tolerance: abs(a - b) <= tol
    // Iterate for max/min of array
    //end
}
// Comparison and clamping

public class Comparison {
    public static void main(String[] args) {
        // Max and min
        System.out.println("Max and min:");
        System.out.println("max(5, 10): " + Math.max(5, 10));
        System.out.println("min(5, 10): " + Math.min(5, 10));
        System.out.println("max(-5, -10): " + Math.max(-5, -10));
        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));

        // Max/min of multiple values
        System.out.println("\nMax/min of multiple:");
        int[] numbers = {5, 2, 9, 1, 7, 3};
        int maxValue = findMax(numbers);
        int minValue = findMin(numbers);
        System.out.println("Array: " + java.util.Arrays.toString(numbers));
        System.out.println("Max: " + maxValue);
        System.out.println("Min: " + minValue);

        // Clamp (restrict to range)
        System.out.println("\nClamp to range [0, 100]:");
        System.out.println("clamp(-10): " + clamp(-10, 0, 100));
        System.out.println("clamp(50): " + clamp(50, 0, 100));
        System.out.println("clamp(150): " + clamp(150, 0, 100));

        // Sign comparison
        System.out.println("\nSign comparison:");
        System.out.println("signum(5): " + Math.signum(5));
        System.out.println("signum(-5): " + Math.signum(-5));
        System.out.println("signum(0): " + Math.signum(0));

        // Same sign check
        System.out.println("\nSame sign:");
        System.out.println("5 and 10: " + sameSign(5, 10));
        System.out.println("5 and -10: " + sameSign(5, -10));
        System.out.println("-5 and -10: " + sameSign(-5, -10));

        // Absolute difference
        System.out.println("\nAbsolute difference:");
        System.out.println("|5 - 10|: " + Math.abs(5 - 10));
        System.out.println("|10 - 5|: " + Math.abs(10 - 5));
        System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));

        // Within tolerance
        System.out.println("\nWithin tolerance:");
        double a = 3.14159;
        double b = 3.14;
        double tolerance = 0.01;
        System.out.println(a + " ≈ " + b + " (±" + tolerance + "): " +
                          withinTolerance(a, b, tolerance));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Keep score in bounds
        int score = 50;
        int bounded = clamp(score, 0, 100);
        System.out.println("Score " + score + " bounded to [0,100]: " + bounded);

        // Volume control
        double volume = 1.5;
        double validVolume = clamp(volume, 0.0, 1.0);
        System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);

        // Temperature range
        int temp = -5;
        int minTemp = 0;
        int maxTemp = 30;
        int validTemp = clamp(temp, minTemp, maxTemp);
        System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);

        // Find range
        int[] values = {23, 45, 12, 67, 34};
        int min = findMin(values);
        int max = findMax(values);
        int range = max - min;
        System.out.println("\nValues: " + java.util.Arrays.toString(values));
        System.out.println("Range: " + min + " to " + max + " (span: " + range + ")");

        // Midpoint
        System.out.println("\nMidpoint:");
        System.out.println("Between 10 and 20: " + midpoint(10, 20));
        System.out.println("Between -5 and 15: " + midpoint(-5, 15));
    }
    public static int findMax(int[] arr) {
        int max = arr[0];
        for (int val : arr) {
            max = Math.max(max, val);
        }
        return max;
    }
    public static int findMin(int[] arr) {
        int min = arr[0];
        for (int val : arr) {
            min = Math.min(min, val);
        }
        return min;
    }
    public static int clamp(int value, int min, int max) {
        return Math.max(min, Math.min(max, value));
    }

    public static double clamp(double value, double min, double max) {
        return Math.max(min, Math.min(max, value));
    }
    public static boolean sameSign(double a, double b) {
        return Math.signum(a) == Math.signum(b);
    }
    public static boolean withinTolerance(double a, double b, double tolerance) {
        return Math.abs(a - b) <= tolerance;
    }
    public static double midpoint(double a, double b) {
        return (a + b) / 2;
    }

    //help h1
    // Math.max(a, b), Math.min(a, b)
    // Clamp: max(min, min(max, value))
    // Math.signum(x) - sign of number
    // Math.abs(a - b) - absolute difference
    // Within tolerance: abs(a - b) <= tol
    // Iterate for max/min of array
    //end
}

                        
1public static void main(String[] args)⋯3public class Comparison {4    public static void main(String[] args) {5        // Max and min6        System.out.println("Max and min:");7        System.out.println("max(5, 10): " + Math.max(5, 10));8        System.out.println("min(5, 10): " + Math.min(5, 10));9        System.out.println("max(-5, -10): " + Math.max(-5, -10));10        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));1112        // Max/min of multiple values13        System.out.println("\nMax/min of multiple:");14        int[] numbers = {5, 2, 9, 1, 7, 3};15        int maxValue = findMax(numbers);16        int minValue = findMin(numbers);⋯outputMax and min:
max(5, 10): 10
min(5, 10): 5
max(-5, -10): -5
min(3.14, 2.71): 2.71

Max/min of multiple:
2max ← 5pass 1 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 5 = arr[0]5;88    for (int val : arr) {⋯
3max ← 5pass 1 of 11⋯87int max = arr[0];88for (int val5 : arr) {89    max→ 5 = Math.max(max, val5);90}⋯All 11 passes — pass 1 is the card abovepassvalmax
155
225
395 → 9
419
579
639
72323
84523 → 45
91245
106745 → 67
113467
4return max;⋯90    }91    return max9;92}⋯
5maxValue ← 9⋯14int[] numbers = {5, 2, 9, 1, 7, 3};15int maxValue→ 9 = findMax(numbers);16int minValue = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));⋯
6min ← 5pass 1 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 5 = arr[0]5;95    for (int val : arr) {⋯
7min ← 5pass 1 of 11⋯94int min = arr[0];95for (int val5 : arr) {96    min→ 5 = Math.min(min, val5);97}⋯All 11 passes — pass 1 is the card abovepassvalmin
155
225 → 2
392
412 → 1
571
631
72323
84523
91223 → 12
106712
113412
8return min;⋯97    }98    return min1;99}⋯
9minValue ← 1⋯15int maxValue = findMax(numbers);16int minValue→ 1 = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));18System.out.println("Max: " + maxValue9);19System.out.println("Min: " + minValue1);2021// Clamp (restrict to range)22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));⋯outputArray: [5, 2, 9, 1, 7, 3]
Max: 9
Min: 1

Clamp to range [0, 100]:
10public static int clamp(int value, int min, int max)pass 1 of 5⋯99}100public static int clamp(int value-10, int min0, int max100) {101    return Math.max(min0, Math.min(max100, value-10));102}⋯All 5 passes — pass 1 is the card abovepassvaluemax
1-10100
250100
3150100
4150100
5-530
11System.out.println("clamp(-10): " + clamp(-10, 0, 100));⋯22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(-10): 0
12System.out.println("clamp(50): " + clamp(50, 0, 100));⋯23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(50): 50
13System.out.println("clamp(150): " + clamp(150, 0, 100));⋯24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));2627// Sign comparison28System.out.println("\nSign comparison:");29System.out.println("signum(5): " + Math.signum(5));30System.out.println("signum(-5): " + Math.signum(-5));31System.out.println("signum(0): " + Math.signum(0));3233// Same sign check34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));⋯outputclamp(150): 100

Sign comparison:
signum(5): 1.0
signum(-5): -1.0
signum(0): 0.0

Same sign:
14public static boolean sameSign(double a, double b)pass 1 of 3⋯106}107public static boolean sameSign(double a5.0, double b10.0) {108    return Math.signum(a5.0) == Math.signum(b10.0);109}⋯All 3 passes — pass 1 is the card abovepassab
15.010.0
25.0-10.0
3-5.0-10.0
15System.out.println("5 and 10: " + sameSign(5, 10));⋯34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and 10: true
16System.out.println("5 and -10: " + sameSign(5, -10));⋯35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and -10: false
17a ← 3.14159, b ← 3.14, tolerance ← 0.01⋯36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));3839// Absolute difference40System.out.println("\nAbsolute difference:");41System.out.println("|5 - 10|: " + Math.abs(5 - 10));42System.out.println("|10 - 5|: " + Math.abs(10 - 5));43System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));4445// Within tolerance46System.out.println("\nWithin tolerance:");47double a→ 3.14159 = 3.14159;48double b→ 3.14 = 3.14;49double tolerance→ 0.01 = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));⋯output-5 and -10: true

Absolute difference:
|5 - 10|: 5
|10 - 5|: 5
|-5 - (-10)|: 5

Within tolerance:
18public static boolean withinTolerance(double a, double b, double toler…⋯109}110public static boolean withinTolerance(double a3.14159, double b3.14, double tolerance0.01) {111    return Math.abs(a3.14159 - b3.14) <= tolerance0.01;112}⋯
19score ← 150⋯49double tolerance = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));5253// Practical examples54System.out.println("\nPractical examples:");5556// Keep score in bounds57int score→ 150 = 150; //@score=-10, 5058int bounded = clamp(score150, 0, 100);59System.out.println("Score " + score + " bounded to [0,100]: " + bounded);⋯output3.14159 ≈ 3.14 (±0.01): true

Practical examples:
20bounded ← 100, volume ← 1.5⋯57int score = 150; //@score=-10, 5058int bounded→ 100 = clamp(score150, 0, 100);59System.out.println("Score " + score150 + " bounded to [0,100]: " + bounded100);6061// Volume control62double volume→ 1.5 = 1.5;63double validVolume = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);⋯outputScore 150 bounded to [0,100]: 100
21public static double clamp(double value, double min, double max)⋯104public static double clamp(double value1.5, double min0.0, double max1.0) {105    return Math.max(min0.0, Math.min(max1.0, value1.5));106}⋯
22validVolume ← 1.0, temp ← -5, minTemp ← 0, maxTemp ← 30⋯62double volume = 1.5;63double validVolume→ 1.0 = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume1.5 + " clamped to [0,1]: " + validVolume1.0);6566// Temperature range67int temp→ -5 = -5;68int minTemp→ 0 = 0;69int maxTemp→ 30 = 30;70int validTemp = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);⋯outputVolume 1.5 clamped to [0,1]: 1.0
23validTemp ← 0⋯69int maxTemp = 30;70int validTemp→ 0 = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp-5 + "° bounded to [" + minTemp0 + "," + maxTemp30 + "]: " + validTemp0);7273// Find range74int[] values = {23, 45, 12, 67, 34};75int min = findMin(values);76int max = findMax(values);⋯outputTemp -5° bounded to [0,30]: 0
24min ← 23pass 2 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 23 = arr[0]23;95    for (int val : arr) {⋯
25return min;⋯97    }98    return min12;99}⋯
26min ← 12⋯74int[] values = {23, 45, 12, 67, 34};75int min→ 12 = findMin(values);76int max = findMax(values);77int range = max - min;⋯
27max ← 23pass 2 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 23 = arr[0]23;88    for (int val : arr) {⋯
28return max;⋯90    }91    return max67;92}⋯
29max ← 67, range ← 55⋯75int min = findMin(values);76int max→ 67 = findMax(values);77int range→ 55 = max67 - min12;78System.out.println("\nValues: " + java.util.Arrays.toString(values));79System.out.println("Range: " + min12 + " to " + max67 + " (span: " + range55 + ")");8081// Midpoint82System.out.println("\nMidpoint:");83System.out.println("Between 10 and 20: " + midpoint(10, 20));84System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯output
Values: [23, 45, 12, 67, 34]
Range: 12 to 67 (span: 55)

Midpoint:
30public static double midpoint(double a, double b)pass 1 of 2⋯112}113public static double midpoint(double a10.0, double b20.0) {114    return (a10.0 + b20.0) / 2;115}⋯
31System.out.println("Between 10 and 20: " + midpoint(10, 20));⋯82    System.out.println("\nMidpoint:");83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween 10 and 20: 15.0
32public static double midpoint(double a, double b)pass 2 of 2⋯112}113public static double midpoint(double a-5.0, double b15.0) {114    return (a-5.0 + b15.0) / 2;115}⋯
33System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween -5 and 15: 5.0
1public static void main(String[] args)⋯3public class Comparison {4    public static void main(String[] args) {5        // Max and min6        System.out.println("Max and min:");7        System.out.println("max(5, 10): " + Math.max(5, 10));8        System.out.println("min(5, 10): " + Math.min(5, 10));9        System.out.println("max(-5, -10): " + Math.max(-5, -10));10        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));1112        // Max/min of multiple values13        System.out.println("\nMax/min of multiple:");14        int[] numbers = {5, 2, 9, 1, 7, 3};15        int maxValue = findMax(numbers);16        int minValue = findMin(numbers);⋯outputMax and min:
max(5, 10): 10
min(5, 10): 5
max(-5, -10): -5
min(3.14, 2.71): 2.71

Max/min of multiple:
2max ← 5pass 1 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 5 = arr[0]5;88    for (int val : arr) {⋯
3max ← 5pass 1 of 11⋯87int max = arr[0];88for (int val5 : arr) {89    max→ 5 = Math.max(max, val5);90}⋯All 11 passes — pass 1 is the card abovepassvalmax
155
225
395 → 9
419
579
639
72323
84523 → 45
91245
106745 → 67
113467
4return max;⋯90    }91    return max9;92}⋯
5maxValue ← 9⋯14int[] numbers = {5, 2, 9, 1, 7, 3};15int maxValue→ 9 = findMax(numbers);16int minValue = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));⋯
6min ← 5pass 1 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 5 = arr[0]5;95    for (int val : arr) {⋯
7min ← 5pass 1 of 11⋯94int min = arr[0];95for (int val5 : arr) {96    min→ 5 = Math.min(min, val5);97}⋯All 11 passes — pass 1 is the card abovepassvalmin
155
225 → 2
392
412 → 1
571
631
72323
84523
91223 → 12
106712
113412
8return min;⋯97    }98    return min1;99}⋯
9minValue ← 1⋯15int maxValue = findMax(numbers);16int minValue→ 1 = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));18System.out.println("Max: " + maxValue9);19System.out.println("Min: " + minValue1);2021// Clamp (restrict to range)22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));⋯outputArray: [5, 2, 9, 1, 7, 3]
Max: 9
Min: 1

Clamp to range [0, 100]:
10public static int clamp(int value, int min, int max)pass 1 of 5⋯99}100public static int clamp(int value-10, int min0, int max100) {101    return Math.max(min0, Math.min(max100, value-10));102}⋯All 5 passes — pass 1 is the card abovepassvaluemax
1-10100
250100
3150100
4-10100
5-530
11System.out.println("clamp(-10): " + clamp(-10, 0, 100));⋯22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(-10): 0
12System.out.println("clamp(50): " + clamp(50, 0, 100));⋯23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(50): 50
13System.out.println("clamp(150): " + clamp(150, 0, 100));⋯24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));2627// Sign comparison28System.out.println("\nSign comparison:");29System.out.println("signum(5): " + Math.signum(5));30System.out.println("signum(-5): " + Math.signum(-5));31System.out.println("signum(0): " + Math.signum(0));3233// Same sign check34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));⋯outputclamp(150): 100

Sign comparison:
signum(5): 1.0
signum(-5): -1.0
signum(0): 0.0

Same sign:
14public static boolean sameSign(double a, double b)pass 1 of 3⋯106}107public static boolean sameSign(double a5.0, double b10.0) {108    return Math.signum(a5.0) == Math.signum(b10.0);109}⋯All 3 passes — pass 1 is the card abovepassab
15.010.0
25.0-10.0
3-5.0-10.0
15System.out.println("5 and 10: " + sameSign(5, 10));⋯34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and 10: true
16System.out.println("5 and -10: " + sameSign(5, -10));⋯35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and -10: false
17a ← 3.14159, b ← 3.14, tolerance ← 0.01⋯36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));3839// Absolute difference40System.out.println("\nAbsolute difference:");41System.out.println("|5 - 10|: " + Math.abs(5 - 10));42System.out.println("|10 - 5|: " + Math.abs(10 - 5));43System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));4445// Within tolerance46System.out.println("\nWithin tolerance:");47double a→ 3.14159 = 3.14159;48double b→ 3.14 = 3.14;49double tolerance→ 0.01 = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));⋯output-5 and -10: true

Absolute difference:
|5 - 10|: 5
|10 - 5|: 5
|-5 - (-10)|: 5

Within tolerance:
18public static boolean withinTolerance(double a, double b, double toler…⋯109}110public static boolean withinTolerance(double a3.14159, double b3.14, double tolerance0.01) {111    return Math.abs(a3.14159 - b3.14) <= tolerance0.01;112}⋯
19score ← -10⋯49double tolerance = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));5253// Practical examples54System.out.println("\nPractical examples:");5556// Keep score in bounds57int score→ -10 = -10;58int bounded = clamp(score-10, 0, 100);59System.out.println("Score " + score + " bounded to [0,100]: " + bounded);⋯output3.14159 ≈ 3.14 (±0.01): true

Practical examples:
20bounded ← 0, volume ← 1.5⋯57int score = -10;58int bounded→ 0 = clamp(score-10, 0, 100);59System.out.println("Score " + score-10 + " bounded to [0,100]: " + bounded0);6061// Volume control62double volume→ 1.5 = 1.5;63double validVolume = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);⋯outputScore -10 bounded to [0,100]: 0
21public static double clamp(double value, double min, double max)⋯104public static double clamp(double value1.5, double min0.0, double max1.0) {105    return Math.max(min0.0, Math.min(max1.0, value1.5));106}⋯
22validVolume ← 1.0, temp ← -5, minTemp ← 0, maxTemp ← 30⋯62double volume = 1.5;63double validVolume→ 1.0 = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume1.5 + " clamped to [0,1]: " + validVolume1.0);6566// Temperature range67int temp→ -5 = -5;68int minTemp→ 0 = 0;69int maxTemp→ 30 = 30;70int validTemp = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);⋯outputVolume 1.5 clamped to [0,1]: 1.0
23validTemp ← 0⋯69int maxTemp = 30;70int validTemp→ 0 = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp-5 + "° bounded to [" + minTemp0 + "," + maxTemp30 + "]: " + validTemp0);7273// Find range74int[] values = {23, 45, 12, 67, 34};75int min = findMin(values);76int max = findMax(values);⋯outputTemp -5° bounded to [0,30]: 0
24min ← 23pass 2 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 23 = arr[0]23;95    for (int val : arr) {⋯
25return min;⋯97    }98    return min12;99}⋯
26min ← 12⋯74int[] values = {23, 45, 12, 67, 34};75int min→ 12 = findMin(values);76int max = findMax(values);77int range = max - min;⋯
27max ← 23pass 2 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 23 = arr[0]23;88    for (int val : arr) {⋯
28return max;⋯90    }91    return max67;92}⋯
29max ← 67, range ← 55⋯75int min = findMin(values);76int max→ 67 = findMax(values);77int range→ 55 = max67 - min12;78System.out.println("\nValues: " + java.util.Arrays.toString(values));79System.out.println("Range: " + min12 + " to " + max67 + " (span: " + range55 + ")");8081// Midpoint82System.out.println("\nMidpoint:");83System.out.println("Between 10 and 20: " + midpoint(10, 20));84System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯output
Values: [23, 45, 12, 67, 34]
Range: 12 to 67 (span: 55)

Midpoint:
30public static double midpoint(double a, double b)pass 1 of 2⋯112}113public static double midpoint(double a10.0, double b20.0) {114    return (a10.0 + b20.0) / 2;115}⋯
31System.out.println("Between 10 and 20: " + midpoint(10, 20));⋯82    System.out.println("\nMidpoint:");83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween 10 and 20: 15.0
32public static double midpoint(double a, double b)pass 2 of 2⋯112}113public static double midpoint(double a-5.0, double b15.0) {114    return (a-5.0 + b15.0) / 2;115}⋯
33System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween -5 and 15: 5.0
1public static void main(String[] args)⋯3public class Comparison {4    public static void main(String[] args) {5        // Max and min6        System.out.println("Max and min:");7        System.out.println("max(5, 10): " + Math.max(5, 10));8        System.out.println("min(5, 10): " + Math.min(5, 10));9        System.out.println("max(-5, -10): " + Math.max(-5, -10));10        System.out.println("min(3.14, 2.71): " + Math.min(3.14, 2.71));1112        // Max/min of multiple values13        System.out.println("\nMax/min of multiple:");14        int[] numbers = {5, 2, 9, 1, 7, 3};15        int maxValue = findMax(numbers);16        int minValue = findMin(numbers);⋯outputMax and min:
max(5, 10): 10
min(5, 10): 5
max(-5, -10): -5
min(3.14, 2.71): 2.71

Max/min of multiple:
2max ← 5pass 1 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 5 = arr[0]5;88    for (int val : arr) {⋯
3max ← 5pass 1 of 11⋯87int max = arr[0];88for (int val5 : arr) {89    max→ 5 = Math.max(max, val5);90}⋯All 11 passes — pass 1 is the card abovepassvalmax
155
225
395 → 9
419
579
639
72323
84523 → 45
91245
106745 → 67
113467
4return max;⋯90    }91    return max9;92}⋯
5maxValue ← 9⋯14int[] numbers = {5, 2, 9, 1, 7, 3};15int maxValue→ 9 = findMax(numbers);16int minValue = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));⋯
6min ← 5pass 1 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 5 = arr[0]5;95    for (int val : arr) {⋯
7min ← 5pass 1 of 11⋯94int min = arr[0];95for (int val5 : arr) {96    min→ 5 = Math.min(min, val5);97}⋯All 11 passes — pass 1 is the card abovepassvalmin
155
225 → 2
392
412 → 1
571
631
72323
84523
91223 → 12
106712
113412
8return min;⋯97    }98    return min1;99}⋯
9minValue ← 1⋯15int maxValue = findMax(numbers);16int minValue→ 1 = findMin(numbers);17System.out.println("Array: " + java.util.Arrays.toString(numbers));18System.out.println("Max: " + maxValue9);19System.out.println("Min: " + minValue1);2021// Clamp (restrict to range)22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));⋯outputArray: [5, 2, 9, 1, 7, 3]
Max: 9
Min: 1

Clamp to range [0, 100]:
10public static int clamp(int value, int min, int max)pass 1 of 5⋯99}100public static int clamp(int value-10, int min0, int max100) {101    return Math.max(min0, Math.min(max100, value-10));102}⋯All 5 passes — pass 1 is the card abovepassvaluemax
1-10100
250100
3150100
450100
5-530
11System.out.println("clamp(-10): " + clamp(-10, 0, 100));⋯22System.out.println("\nClamp to range [0, 100]:");23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(-10): 0
12System.out.println("clamp(50): " + clamp(50, 0, 100));⋯23System.out.println("clamp(-10): " + clamp(-10, 0, 100));24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));⋯outputclamp(50): 50
13System.out.println("clamp(150): " + clamp(150, 0, 100));⋯24System.out.println("clamp(50): " + clamp(50, 0, 100));25System.out.println("clamp(150): " + clamp(150, 0, 100));2627// Sign comparison28System.out.println("\nSign comparison:");29System.out.println("signum(5): " + Math.signum(5));30System.out.println("signum(-5): " + Math.signum(-5));31System.out.println("signum(0): " + Math.signum(0));3233// Same sign check34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));⋯outputclamp(150): 100

Sign comparison:
signum(5): 1.0
signum(-5): -1.0
signum(0): 0.0

Same sign:
14public static boolean sameSign(double a, double b)pass 1 of 3⋯106}107public static boolean sameSign(double a5.0, double b10.0) {108    return Math.signum(a5.0) == Math.signum(b10.0);109}⋯All 3 passes — pass 1 is the card abovepassab
15.010.0
25.0-10.0
3-5.0-10.0
15System.out.println("5 and 10: " + sameSign(5, 10));⋯34System.out.println("\nSame sign:");35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and 10: true
16System.out.println("5 and -10: " + sameSign(5, -10));⋯35System.out.println("5 and 10: " + sameSign(5, 10));36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));⋯output5 and -10: false
17a ← 3.14159, b ← 3.14, tolerance ← 0.01⋯36System.out.println("5 and -10: " + sameSign(5, -10));37System.out.println("-5 and -10: " + sameSign(-5, -10));3839// Absolute difference40System.out.println("\nAbsolute difference:");41System.out.println("|5 - 10|: " + Math.abs(5 - 10));42System.out.println("|10 - 5|: " + Math.abs(10 - 5));43System.out.println("|-5 - (-10)|: " + Math.abs(-5 - (-10)));4445// Within tolerance46System.out.println("\nWithin tolerance:");47double a→ 3.14159 = 3.14159;48double b→ 3.14 = 3.14;49double tolerance→ 0.01 = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));⋯output-5 and -10: true

Absolute difference:
|5 - 10|: 5
|10 - 5|: 5
|-5 - (-10)|: 5

Within tolerance:
18public static boolean withinTolerance(double a, double b, double toler…⋯109}110public static boolean withinTolerance(double a3.14159, double b3.14, double tolerance0.01) {111    return Math.abs(a3.14159 - b3.14) <= tolerance0.01;112}⋯
19score ← 50⋯49double tolerance = 0.01;50System.out.println(a3.14159 + " ≈ " + b3.14 + " (±" + tolerance0.01 + "): " + 51                  withinTolerance(a3.14159, b3.14, tolerance0.01));5253// Practical examples54System.out.println("\nPractical examples:");5556// Keep score in bounds57int score→ 50 = 50;58int bounded = clamp(score50, 0, 100);59System.out.println("Score " + score + " bounded to [0,100]: " + bounded);⋯output3.14159 ≈ 3.14 (±0.01): true

Practical examples:
20bounded ← 50, volume ← 1.5⋯57int score = 50;58int bounded→ 50 = clamp(score50, 0, 100);59System.out.println("Score " + score50 + " bounded to [0,100]: " + bounded50);6061// Volume control62double volume→ 1.5 = 1.5;63double validVolume = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume + " clamped to [0,1]: " + validVolume);⋯outputScore 50 bounded to [0,100]: 50
21public static double clamp(double value, double min, double max)⋯104public static double clamp(double value1.5, double min0.0, double max1.0) {105    return Math.max(min0.0, Math.min(max1.0, value1.5));106}⋯
22validVolume ← 1.0, temp ← -5, minTemp ← 0, maxTemp ← 30⋯62double volume = 1.5;63double validVolume→ 1.0 = clamp(volume1.5, 0.0, 1.0);64System.out.println("Volume " + volume1.5 + " clamped to [0,1]: " + validVolume1.0);6566// Temperature range67int temp→ -5 = -5;68int minTemp→ 0 = 0;69int maxTemp→ 30 = 30;70int validTemp = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp + "° bounded to [" + minTemp + "," + maxTemp + "]: " + validTemp);⋯outputVolume 1.5 clamped to [0,1]: 1.0
23validTemp ← 0⋯69int maxTemp = 30;70int validTemp→ 0 = clamp(temp-5, minTemp0, maxTemp30);71System.out.println("Temp " + temp-5 + "° bounded to [" + minTemp0 + "," + maxTemp30 + "]: " + validTemp0);7273// Find range74int[] values = {23, 45, 12, 67, 34};75int min = findMin(values);76int max = findMax(values);⋯outputTemp -5° bounded to [0,30]: 0
24min ← 23pass 2 of 2⋯92}93public static int findMin(int[] arr) {94    int min→ 23 = arr[0]23;95    for (int val : arr) {⋯
25return min;⋯97    }98    return min12;99}⋯
26min ← 12⋯74int[] values = {23, 45, 12, 67, 34};75int min→ 12 = findMin(values);76int max = findMax(values);77int range = max - min;⋯
27max ← 23pass 2 of 2⋯85}86public static int findMax(int[] arr) {87    int max→ 23 = arr[0]23;88    for (int val : arr) {⋯
28return max;⋯90    }91    return max67;92}⋯
29max ← 67, range ← 55⋯75int min = findMin(values);76int max→ 67 = findMax(values);77int range→ 55 = max67 - min12;78System.out.println("\nValues: " + java.util.Arrays.toString(values));79System.out.println("Range: " + min12 + " to " + max67 + " (span: " + range55 + ")");8081// Midpoint82System.out.println("\nMidpoint:");83System.out.println("Between 10 and 20: " + midpoint(10, 20));84System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯output
Values: [23, 45, 12, 67, 34]
Range: 12 to 67 (span: 55)

Midpoint:
30public static double midpoint(double a, double b)pass 1 of 2⋯112}113public static double midpoint(double a10.0, double b20.0) {114    return (a10.0 + b20.0) / 2;115}⋯
31System.out.println("Between 10 and 20: " + midpoint(10, 20));⋯82    System.out.println("\nMidpoint:");83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween 10 and 20: 15.0
32public static double midpoint(double a, double b)pass 2 of 2⋯112}113public static double midpoint(double a-5.0, double b15.0) {114    return (a-5.0 + b15.0) / 2;115}⋯
33System.out.println("Between -5 and 15: " + midpoint(-5, 15));⋯83    System.out.println("Between 10 and 20: " + midpoint(10, 20));84    System.out.println("Between -5 and 15: " + midpoint(-5, 15));85}⋯outputBetween -5 and 15: 5.0

All 11 passes — pass 1 is the card above
pass	`val`	`max`
1	5	5
2	2	5
3	9	5 → 9
4	1	9
5	7	9
6	3	9
7	23	23
8	45	23 → 45
9	12	45
10	67	45 → 67
11	34	67

All 11 passes — pass 1 is the card above
pass	`val`	`min`
1	5	5
2	2	5 → 2
3	9	2
4	1	2 → 1
5	7	1
6	3	1
7	23	23
8	45	23
9	12	23 → 12
10	67	12
11	34	12

All 5 passes — pass 1 is the card above
pass	`value`	`max`
1	-10	100
2	50	100
3	150	100
4	150	100
5	-5	30

All 3 passes — pass 1 is the card above
pass	`a`	`b`
1	5.0	10.0
2	5.0	-10.0
3	-5.0	-10.0

All 11 passes — pass 1 is the card above
pass	`val`	`max`
1	5	5
2	2	5
3	9	5 → 9
4	1	9
5	7	9
6	3	9
7	23	23
8	45	23 → 45
9	12	45
10	67	45 → 67
11	34	67

All 11 passes — pass 1 is the card above
pass	`val`	`min`
1	5	5
2	2	5 → 2
3	9	2
4	1	2 → 1
5	7	1
6	3	1
7	23	23
8	45	23
9	12	23 → 12
10	67	12
11	34	12

All 5 passes — pass 1 is the card above
pass	`value`	`max`
1	-10	100
2	50	100
3	150	100
4	-10	100
5	-5	30

All 3 passes — pass 1 is the card above
pass	`a`	`b`
1	5.0	10.0
2	5.0	-10.0
3	-5.0	-10.0

All 11 passes — pass 1 is the card above
pass	`val`	`max`
1	5	5
2	2	5
3	9	5 → 9
4	1	9
5	7	9
6	3	9
7	23	23
8	45	23 → 45
9	12	45
10	67	45 → 67
11	34	67

All 11 passes — pass 1 is the card above
pass	`val`	`min`
1	5	5
2	2	5 → 2
3	9	2
4	1	2 → 1
5	7	1
6	3	1
7	23	23
8	45	23
9	12	23 → 12
10	67	12
11	34	12

All 5 passes — pass 1 is the card above
pass	`value`	`max`
1	-10	100
2	50	100
3	150	100
4	50	100
5	-5	30

All 3 passes — pass 1 is the card above
pass	`a`	`b`
1	5.0	10.0
2	5.0	-10.0
3	-5.0	-10.0

Trigonometric Functions

Calculate sine, cosine, and other trigonometric values.

                            degrees
                            degrees = 45 (default)degrees = 30degrees = 60
                        

                            Trigonometry.java
                        
Replay: real traced execution (multi-file project)

                            // Trigonometric functions

public class Trigonometry {
    public static void main(String[] args) {
        // Angles in radians
        double angle = Math.PI / 4;  // 45 degrees

        System.out.println("Angle: " + angle + " radians (45 degrees)");
        System.out.println();

        // Basic trig functions
        System.out.println("Basic trigonometry:");
        System.out.println("sin(π/4): " + Math.sin(angle));
        System.out.println("cos(π/4): " + Math.cos(angle));
        System.out.println("tan(π/4): " + Math.tan(angle));

        // Common angles
        System.out.println("\nCommon angles:");
        System.out.println("sin(0): " + Math.sin(0));
        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees
        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees
        System.out.println("cos(0): " + Math.cos(0));
        System.out.println("cos(π): " + Math.cos(Math.PI));

        // Inverse trig functions
        System.out.println("\nInverse trigonometry:");
        System.out.println("asin(1): " + Math.asin(1));            // π/2
        System.out.println("acos(0): " + Math.acos(0));            // π/2
        System.out.println("atan(1): " + Math.atan(1));            // π/4
        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/4

        // Hyperbolic functions
        System.out.println("\nHyperbolic functions:");
        System.out.println("sinh(1): " + Math.sinh(1));
        System.out.println("cosh(1): " + Math.cosh(1));
        System.out.println("tanh(1): " + Math.tanh(1));

        // Degree/radian conversion
        System.out.println("\nDegree/radian conversion:");
        double degrees = 45;
        double radians = Math.toRadians(degrees);
        System.out.println(degrees + " degrees = " + radians + " radians");
        System.out.println(radians + " radians = " + Math.toDegrees(radians) + " degrees");

        // Trig with degrees
        System.out.println("\nTrig with degrees (convert first):");
        double angle45 = Math.toRadians(45);
        double angle90 = Math.toRadians(90);
        System.out.println("sin(45°): " + Math.sin(angle45));
        System.out.println("cos(90°): " + Math.cos(angle90));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Right triangle
        double adjacent = 3;
        double opposite = 4;
        double hypotenuse = Math.sqrt(Math.pow(adjacent, 2) + Math.pow(opposite, 2));
        double angleRad = Math.atan2(opposite, adjacent);
        double angleDeg = Math.toDegrees(angleRad);

        System.out.println("Right triangle (3, 4, ?):");
        System.out.println("Hypotenuse: " + hypotenuse);
        System.out.println("Angle: " + angleDeg + " degrees");

        // Circle point
        double radius = 10;
        double angleCircle = Math.toRadians(60);
        double x = radius * Math.cos(angleCircle);
        double y = radius * Math.sin(angleCircle);
        System.out.println("\nPoint on circle (r=10, 60°):");
        System.out.println("x: " + x);
        System.out.println("y: " + y);

        // Distance and angle between points
        double x1 = 0, y1 = 0;
        double x2 = 3, y2 = 3;
        double distance = Math.sqrt(Math.pow(x2 - x1, 2) + Math.pow(y2 - y1, 2));
        double angleToPoint = Math.toDegrees(Math.atan2(y2 - y1, x2 - x1));

        System.out.println("\nFrom (0,0) to (3,3):");
        System.out.println("Distance: " + distance);
        System.out.println("Angle: " + angleToPoint + " degrees");

        // Table of sine values
        System.out.println("\nSine values (0° to 90°, step 15°):");
        for (int deg = 0; deg <= 90; deg += 15) {
            double rad = Math.toRadians(deg);
            System.out.printf("%3d° : %.4f%n", deg, Math.sin(rad));
        }
    }

    //help h1
    // Math.sin(x), Math.cos(x), Math.tan(x) - trig functions
    // Math.asin(x), Math.acos(x), Math.atan(x) - inverse
    // Math.atan2(y, x) - angle from origin to (x,y)
    // Math.toRadians(degrees) - convert to radians
    // Math.toDegrees(radians) - convert to degrees
    // Angles in radians by default
    // Use atan2 for proper quadrant
    //end
}
// Trigonometric functions

public class Trigonometry {
    public static void main(String[] args) {
        // Angles in radians
        double angle = Math.PI / 4;  // 45 degrees

        System.out.println("Angle: " + angle + " radians (45 degrees)");
        System.out.println();

        // Basic trig functions
        System.out.println("Basic trigonometry:");
        System.out.println("sin(π/4): " + Math.sin(angle));
        System.out.println("cos(π/4): " + Math.cos(angle));
        System.out.println("tan(π/4): " + Math.tan(angle));

        // Common angles
        System.out.println("\nCommon angles:");
        System.out.println("sin(0): " + Math.sin(0));
        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees
        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees
        System.out.println("cos(0): " + Math.cos(0));
        System.out.println("cos(π): " + Math.cos(Math.PI));

        // Inverse trig functions
        System.out.println("\nInverse trigonometry:");
        System.out.println("asin(1): " + Math.asin(1));            // π/2
        System.out.println("acos(0): " + Math.acos(0));            // π/2
        System.out.println("atan(1): " + Math.atan(1));            // π/4
        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/4

        // Hyperbolic functions
        System.out.println("\nHyperbolic functions:");
        System.out.println("sinh(1): " + Math.sinh(1));
        System.out.println("cosh(1): " + Math.cosh(1));
        System.out.println("tanh(1): " + Math.tanh(1));

        // Degree/radian conversion
        System.out.println("\nDegree/radian conversion:");
        double degrees = 30;
        double radians = Math.toRadians(degrees);
        System.out.println(degrees + " degrees = " + radians + " radians");
        System.out.println(radians + " radians = " + Math.toDegrees(radians) + " degrees");

        // Trig with degrees
        System.out.println("\nTrig with degrees (convert first):");
        double angle45 = Math.toRadians(45);
        double angle90 = Math.toRadians(90);
        System.out.println("sin(45°): " + Math.sin(angle45));
        System.out.println("cos(90°): " + Math.cos(angle90));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Right triangle
        double adjacent = 3;
        double opposite = 4;
        double hypotenuse = Math.sqrt(Math.pow(adjacent, 2) + Math.pow(opposite, 2));
        double angleRad = Math.atan2(opposite, adjacent);
        double angleDeg = Math.toDegrees(angleRad);

        System.out.println("Right triangle (3, 4, ?):");
        System.out.println("Hypotenuse: " + hypotenuse);
        System.out.println("Angle: " + angleDeg + " degrees");

        // Circle point
        double radius = 10;
        double angleCircle = Math.toRadians(60);
        double x = radius * Math.cos(angleCircle);
        double y = radius * Math.sin(angleCircle);
        System.out.println("\nPoint on circle (r=10, 60°):");
        System.out.println("x: " + x);
        System.out.println("y: " + y);

        // Distance and angle between points
        double x1 = 0, y1 = 0;
        double x2 = 3, y2 = 3;
        double distance = Math.sqrt(Math.pow(x2 - x1, 2) + Math.pow(y2 - y1, 2));
        double angleToPoint = Math.toDegrees(Math.atan2(y2 - y1, x2 - x1));

        System.out.println("\nFrom (0,0) to (3,3):");
        System.out.println("Distance: " + distance);
        System.out.println("Angle: " + angleToPoint + " degrees");

        // Table of sine values
        System.out.println("\nSine values (0° to 90°, step 15°):");
        for (int deg = 0; deg <= 90; deg += 15) {
            double rad = Math.toRadians(deg);
            System.out.printf("%3d° : %.4f%n", deg, Math.sin(rad));
        }
    }

    //help h1
    // Math.sin(x), Math.cos(x), Math.tan(x) - trig functions
    // Math.asin(x), Math.acos(x), Math.atan(x) - inverse
    // Math.atan2(y, x) - angle from origin to (x,y)
    // Math.toRadians(degrees) - convert to radians
    // Math.toDegrees(radians) - convert to degrees
    // Angles in radians by default
    // Use atan2 for proper quadrant
    //end
}
// Trigonometric functions

public class Trigonometry {
    public static void main(String[] args) {
        // Angles in radians
        double angle = Math.PI / 4;  // 45 degrees

        System.out.println("Angle: " + angle + " radians (45 degrees)");
        System.out.println();

        // Basic trig functions
        System.out.println("Basic trigonometry:");
        System.out.println("sin(π/4): " + Math.sin(angle));
        System.out.println("cos(π/4): " + Math.cos(angle));
        System.out.println("tan(π/4): " + Math.tan(angle));

        // Common angles
        System.out.println("\nCommon angles:");
        System.out.println("sin(0): " + Math.sin(0));
        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees
        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees
        System.out.println("cos(0): " + Math.cos(0));
        System.out.println("cos(π): " + Math.cos(Math.PI));

        // Inverse trig functions
        System.out.println("\nInverse trigonometry:");
        System.out.println("asin(1): " + Math.asin(1));            // π/2
        System.out.println("acos(0): " + Math.acos(0));            // π/2
        System.out.println("atan(1): " + Math.atan(1));            // π/4
        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/4

        // Hyperbolic functions
        System.out.println("\nHyperbolic functions:");
        System.out.println("sinh(1): " + Math.sinh(1));
        System.out.println("cosh(1): " + Math.cosh(1));
        System.out.println("tanh(1): " + Math.tanh(1));

        // Degree/radian conversion
        System.out.println("\nDegree/radian conversion:");
        double degrees = 60;
        double radians = Math.toRadians(degrees);
        System.out.println(degrees + " degrees = " + radians + " radians");
        System.out.println(radians + " radians = " + Math.toDegrees(radians) + " degrees");

        // Trig with degrees
        System.out.println("\nTrig with degrees (convert first):");
        double angle45 = Math.toRadians(45);
        double angle90 = Math.toRadians(90);
        System.out.println("sin(45°): " + Math.sin(angle45));
        System.out.println("cos(90°): " + Math.cos(angle90));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Right triangle
        double adjacent = 3;
        double opposite = 4;
        double hypotenuse = Math.sqrt(Math.pow(adjacent, 2) + Math.pow(opposite, 2));
        double angleRad = Math.atan2(opposite, adjacent);
        double angleDeg = Math.toDegrees(angleRad);

        System.out.println("Right triangle (3, 4, ?):");
        System.out.println("Hypotenuse: " + hypotenuse);
        System.out.println("Angle: " + angleDeg + " degrees");

        // Circle point
        double radius = 10;
        double angleCircle = Math.toRadians(60);
        double x = radius * Math.cos(angleCircle);
        double y = radius * Math.sin(angleCircle);
        System.out.println("\nPoint on circle (r=10, 60°):");
        System.out.println("x: " + x);
        System.out.println("y: " + y);

        // Distance and angle between points
        double x1 = 0, y1 = 0;
        double x2 = 3, y2 = 3;
        double distance = Math.sqrt(Math.pow(x2 - x1, 2) + Math.pow(y2 - y1, 2));
        double angleToPoint = Math.toDegrees(Math.atan2(y2 - y1, x2 - x1));

        System.out.println("\nFrom (0,0) to (3,3):");
        System.out.println("Distance: " + distance);
        System.out.println("Angle: " + angleToPoint + " degrees");

        // Table of sine values
        System.out.println("\nSine values (0° to 90°, step 15°):");
        for (int deg = 0; deg <= 90; deg += 15) {
            double rad = Math.toRadians(deg);
            System.out.printf("%3d° : %.4f%n", deg, Math.sin(rad));
        }
    }

    //help h1
    // Math.sin(x), Math.cos(x), Math.tan(x) - trig functions
    // Math.asin(x), Math.acos(x), Math.atan(x) - inverse
    // Math.atan2(y, x) - angle from origin to (x,y)
    // Math.toRadians(degrees) - convert to radians
    // Math.toDegrees(radians) - convert to degrees
    // Angles in radians by default
    // Use atan2 for proper quadrant
    //end
}

                        
1angle ← 0.7853981633974483, degrees ← 45.0, radians ← 0.7853981633974483⋯3public class Trigonometry {4    public static void main(String[] args) {5        // Angles in radians6        double angle→ 0.7853981633974483 = Math.PI / 4;  // 45 degrees7        8        System.out.println("Angle: " + angle0.7853981633974483 + " radians (45 degrees)");9        System.out.println();1011        // Basic trig functions12        System.out.println("Basic trigonometry:");13        System.out.println("sin(π/4): " + Math.sin(angle0.7853981633974483));14        System.out.println("cos(π/4): " + Math.cos(angle0.7853981633974483));15        System.out.println("tan(π/4): " + Math.tan(angle0.7853981633974483));1617        // Common angles18        System.out.println("\nCommon angles:");19        System.out.println("sin(0): " + Math.sin(0));20        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees21        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees22        System.out.println("cos(0): " + Math.cos(0));23        System.out.println("cos(π): " + Math.cos(Math.PI));2425        // Inverse trig functions26        System.out.println("\nInverse trigonometry:");27        System.out.println("asin(1): " + Math.asin(1));            // π/228        System.out.println("acos(0): " + Math.acos(0));            // π/229        System.out.println("atan(1): " + Math.atan(1));            // π/430        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/43132        // Hyperbolic functions33        System.out.println("\nHyperbolic functions:");34        System.out.println("sinh(1): " + Math.sinh(1));35        System.out.println("cosh(1): " + Math.cosh(1));36        System.out.println("tanh(1): " + Math.tanh(1));3738        // Degree/radian conversion39        System.out.println("\nDegree/radian conversion:");40        double degrees→ 45.0 = 45; //@degrees=30, 6041        double radians→ 0.7853981633974483 = Math.toRadians(degrees45.0);42        System.out.println(degrees45.0 + " degrees = " + radians0.7853981633974483 + " radians");43        System.out.println(radians0.7853981633974483 + " radians = " + Math.toDegrees(radians) + " degrees");4445        // Trig with degrees46        System.out.println("\nTrig with degrees (convert first):");47        double angle45→ 0.7853981633974483 = Math.toRadians(45);48        double angle90→ 1.5707963267948966 = Math.toRadians(90);49        System.out.println("sin(45°): " + Math.sin(angle450.7853981633974483));50        System.out.println("cos(90°): " + Math.cos(angle901.5707963267948966));5152        // Practical examples53        System.out.println("\nPractical examples:");54        55        // Right triangle56        double adjacent→ 3.0 = 3;57        double opposite→ 4.0 = 4;58        double hypotenuse→ 5.0 = Math.sqrt(Math.pow(adjacent3.0, 2) + Math.pow(opposite4.0, 2));59        double angleRad→ 0.9272952180016122 = Math.atan2(opposite4.0, adjacent3.0);60        double angleDeg→ 53.13010235415598 = Math.toDegrees(angleRad0.9272952180016122);61        62        System.out.println("Right triangle (3, 4, ?):");63        System.out.println("Hypotenuse: " + hypotenuse5.0);64        System.out.println("Angle: " + angleDeg53.13010235415598 + " degrees");6566        // Circle point67        double radius→ 10.0 = 10;68        double angleCircle→ 1.0471975511965976 = Math.toRadians(60);69        double x→ 5.000000000000001 = radius10.0 * Math.cos(angleCircle1.0471975511965976);70        double y→ 8.660254037844386 = radius10.0 * Math.sin(angleCircle1.0471975511965976);71        System.out.println("\nPoint on circle (r=10, 60°):");72        System.out.println("x: " + x5.000000000000001);73        System.out.println("y: " + y8.660254037844386);7475        // Distance and angle between points76        double x1 = 0, y1→ 0.0 = 0;77        double x2 = 3, y2→ 3.0 = 3;78        double distance→ 4.242640687119285 = Math.sqrt(Math.pow(x23.0 - x10.0, 2) + Math.pow(y23.0 - y10.0, 2));79        double angleToPoint→ 45.0 = Math.toDegrees(Math.atan2(y23.0 - y10.0, x23.0 - x10.0));80        81        System.out.println("\nFrom (0,0) to (3,3):");82        System.out.println("Distance: " + distance4.242640687119285);83        System.out.println("Angle: " + angleToPoint45.0 + " degrees");8485        // Table of sine values86        System.out.println("\nSine values (0° to 90°, step 15°):");87        for (int deg = 0; deg <= 90; deg += 15) {⋯outputAngle: 0.7853981633974483 radians (45 degrees)
Basic trigonometry:
sin(π/4): 0.7071067811865475
cos(π/4): 0.7071067811865476
tan(π/4): 0.9999999999999999

Common angles:
sin(0): 0.0
sin(π/2): 1.0
sin(π): 1.2246467991473532E-16
cos(0): 1.0
cos(π): -1.0

Inverse trigonometry:
asin(1): 1.5707963267948966
acos(0): 1.5707963267948966
atan(1): 0.7853981633974483
atan2(1, 1): 0.7853981633974483

Hyperbolic functions:
sinh(1): 1.1752011936438014
cosh(1): 1.543080634815244
tanh(1): 0.7615941559557649

Degree/radian conversion:
45.0 degrees = 0.7853981633974483 radians
0.7853981633974483 radians = 45.0 degrees

Trig with degrees (convert first):
sin(45°): 0.7071067811865475
cos(90°): 6.123233995736766E-17

Practical examples:
Right triangle (3, 4, ?):
Hypotenuse: 5.0
Angle: 53.13010235415598 degrees

Point on circle (r=10, 60°):
x: 5.000000000000001
y: 8.660254037844386

From (0,0) to (3,3):
Distance: 4.242640687119285
Angle: 45.0 degrees

Sine values (0° to 90°, step 15°):
2rad ← 0.0pass 1 of 7⋯86System.out.println("\nSine values (0° to 90°, step 15°):");87for (int deg0 = 0; deg <= 90; deg += 15) {88    double rad→ 0.0 = Math.toRadians(deg0);89    System.out.printf("%3d° : %.4f%n", deg0, Math.sin(rad0.0));90}⋯All 7 passes — pass 1 is the card abovepassdegrad
100.0
2150.2617993877991494
3300.5235987755982988
4450.7853981633974483
5601.0471975511965976
6751.3089969389957472
7901.5707963267948966
1angle ← 0.7853981633974483, degrees ← 30.0, radians ← 0.5235987755982988⋯3public class Trigonometry {4    public static void main(String[] args) {5        // Angles in radians6        double angle→ 0.7853981633974483 = Math.PI / 4;  // 45 degrees7        8        System.out.println("Angle: " + angle0.7853981633974483 + " radians (45 degrees)");9        System.out.println();1011        // Basic trig functions12        System.out.println("Basic trigonometry:");13        System.out.println("sin(π/4): " + Math.sin(angle0.7853981633974483));14        System.out.println("cos(π/4): " + Math.cos(angle0.7853981633974483));15        System.out.println("tan(π/4): " + Math.tan(angle0.7853981633974483));1617        // Common angles18        System.out.println("\nCommon angles:");19        System.out.println("sin(0): " + Math.sin(0));20        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees21        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees22        System.out.println("cos(0): " + Math.cos(0));23        System.out.println("cos(π): " + Math.cos(Math.PI));2425        // Inverse trig functions26        System.out.println("\nInverse trigonometry:");27        System.out.println("asin(1): " + Math.asin(1));            // π/228        System.out.println("acos(0): " + Math.acos(0));            // π/229        System.out.println("atan(1): " + Math.atan(1));            // π/430        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/43132        // Hyperbolic functions33        System.out.println("\nHyperbolic functions:");34        System.out.println("sinh(1): " + Math.sinh(1));35        System.out.println("cosh(1): " + Math.cosh(1));36        System.out.println("tanh(1): " + Math.tanh(1));3738        // Degree/radian conversion39        System.out.println("\nDegree/radian conversion:");40        double degrees→ 30.0 = 30;41        double radians→ 0.5235987755982988 = Math.toRadians(degrees30.0);42        System.out.println(degrees30.0 + " degrees = " + radians0.5235987755982988 + " radians");43        System.out.println(radians0.5235987755982988 + " radians = " + Math.toDegrees(radians) + " degrees");4445        // Trig with degrees46        System.out.println("\nTrig with degrees (convert first):");47        double angle45→ 0.7853981633974483 = Math.toRadians(45);48        double angle90→ 1.5707963267948966 = Math.toRadians(90);49        System.out.println("sin(45°): " + Math.sin(angle450.7853981633974483));50        System.out.println("cos(90°): " + Math.cos(angle901.5707963267948966));5152        // Practical examples53        System.out.println("\nPractical examples:");54        55        // Right triangle56        double adjacent→ 3.0 = 3;57        double opposite→ 4.0 = 4;58        double hypotenuse→ 5.0 = Math.sqrt(Math.pow(adjacent3.0, 2) + Math.pow(opposite4.0, 2));59        double angleRad→ 0.9272952180016122 = Math.atan2(opposite4.0, adjacent3.0);60        double angleDeg→ 53.13010235415598 = Math.toDegrees(angleRad0.9272952180016122);61        62        System.out.println("Right triangle (3, 4, ?):");63        System.out.println("Hypotenuse: " + hypotenuse5.0);64        System.out.println("Angle: " + angleDeg53.13010235415598 + " degrees");6566        // Circle point67        double radius→ 10.0 = 10;68        double angleCircle→ 1.0471975511965976 = Math.toRadians(60);69        double x→ 5.000000000000001 = radius10.0 * Math.cos(angleCircle1.0471975511965976);70        double y→ 8.660254037844386 = radius10.0 * Math.sin(angleCircle1.0471975511965976);71        System.out.println("\nPoint on circle (r=10, 60°):");72        System.out.println("x: " + x5.000000000000001);73        System.out.println("y: " + y8.660254037844386);7475        // Distance and angle between points76        double x1 = 0, y1→ 0.0 = 0;77        double x2 = 3, y2→ 3.0 = 3;78        double distance→ 4.242640687119285 = Math.sqrt(Math.pow(x23.0 - x10.0, 2) + Math.pow(y23.0 - y10.0, 2));79        double angleToPoint→ 45.0 = Math.toDegrees(Math.atan2(y23.0 - y10.0, x23.0 - x10.0));80        81        System.out.println("\nFrom (0,0) to (3,3):");82        System.out.println("Distance: " + distance4.242640687119285);83        System.out.println("Angle: " + angleToPoint45.0 + " degrees");8485        // Table of sine values86        System.out.println("\nSine values (0° to 90°, step 15°):");87        for (int deg = 0; deg <= 90; deg += 15) {⋯outputAngle: 0.7853981633974483 radians (45 degrees)
Basic trigonometry:
sin(π/4): 0.7071067811865475
cos(π/4): 0.7071067811865476
tan(π/4): 0.9999999999999999

Common angles:
sin(0): 0.0
sin(π/2): 1.0
sin(π): 1.2246467991473532E-16
cos(0): 1.0
cos(π): -1.0

Inverse trigonometry:
asin(1): 1.5707963267948966
acos(0): 1.5707963267948966
atan(1): 0.7853981633974483
atan2(1, 1): 0.7853981633974483

Hyperbolic functions:
sinh(1): 1.1752011936438014
cosh(1): 1.543080634815244
tanh(1): 0.7615941559557649

Degree/radian conversion:
30.0 degrees = 0.5235987755982988 radians
0.5235987755982988 radians = 29.999999999999996 degrees

Trig with degrees (convert first):
sin(45°): 0.7071067811865475
cos(90°): 6.123233995736766E-17

Practical examples:
Right triangle (3, 4, ?):
Hypotenuse: 5.0
Angle: 53.13010235415598 degrees

Point on circle (r=10, 60°):
x: 5.000000000000001
y: 8.660254037844386

From (0,0) to (3,3):
Distance: 4.242640687119285
Angle: 45.0 degrees

Sine values (0° to 90°, step 15°):
2rad ← 0.0pass 1 of 7⋯86System.out.println("\nSine values (0° to 90°, step 15°):");87for (int deg0 = 0; deg <= 90; deg += 15) {88    double rad→ 0.0 = Math.toRadians(deg0);89    System.out.printf("%3d° : %.4f%n", deg0, Math.sin(rad0.0));90}⋯All 7 passes — pass 1 is the card abovepassdegrad
100.0
2150.2617993877991494
3300.5235987755982988
4450.7853981633974483
5601.0471975511965976
6751.3089969389957472
7901.5707963267948966
1angle ← 0.7853981633974483, degrees ← 60.0, radians ← 1.0471975511965976⋯3public class Trigonometry {4    public static void main(String[] args) {5        // Angles in radians6        double angle→ 0.7853981633974483 = Math.PI / 4;  // 45 degrees7        8        System.out.println("Angle: " + angle0.7853981633974483 + " radians (45 degrees)");9        System.out.println();1011        // Basic trig functions12        System.out.println("Basic trigonometry:");13        System.out.println("sin(π/4): " + Math.sin(angle0.7853981633974483));14        System.out.println("cos(π/4): " + Math.cos(angle0.7853981633974483));15        System.out.println("tan(π/4): " + Math.tan(angle0.7853981633974483));1617        // Common angles18        System.out.println("\nCommon angles:");19        System.out.println("sin(0): " + Math.sin(0));20        System.out.println("sin(π/2): " + Math.sin(Math.PI / 2));  // 90 degrees21        System.out.println("sin(π): " + Math.sin(Math.PI));        // 180 degrees22        System.out.println("cos(0): " + Math.cos(0));23        System.out.println("cos(π): " + Math.cos(Math.PI));2425        // Inverse trig functions26        System.out.println("\nInverse trigonometry:");27        System.out.println("asin(1): " + Math.asin(1));            // π/228        System.out.println("acos(0): " + Math.acos(0));            // π/229        System.out.println("atan(1): " + Math.atan(1));            // π/430        System.out.println("atan2(1, 1): " + Math.atan2(1, 1));    // π/43132        // Hyperbolic functions33        System.out.println("\nHyperbolic functions:");34        System.out.println("sinh(1): " + Math.sinh(1));35        System.out.println("cosh(1): " + Math.cosh(1));36        System.out.println("tanh(1): " + Math.tanh(1));3738        // Degree/radian conversion39        System.out.println("\nDegree/radian conversion:");40        double degrees→ 60.0 = 60;41        double radians→ 1.0471975511965976 = Math.toRadians(degrees60.0);42        System.out.println(degrees60.0 + " degrees = " + radians1.0471975511965976 + " radians");43        System.out.println(radians1.0471975511965976 + " radians = " + Math.toDegrees(radians) + " degrees");4445        // Trig with degrees46        System.out.println("\nTrig with degrees (convert first):");47        double angle45→ 0.7853981633974483 = Math.toRadians(45);48        double angle90→ 1.5707963267948966 = Math.toRadians(90);49        System.out.println("sin(45°): " + Math.sin(angle450.7853981633974483));50        System.out.println("cos(90°): " + Math.cos(angle901.5707963267948966));5152        // Practical examples53        System.out.println("\nPractical examples:");54        55        // Right triangle56        double adjacent→ 3.0 = 3;57        double opposite→ 4.0 = 4;58        double hypotenuse→ 5.0 = Math.sqrt(Math.pow(adjacent3.0, 2) + Math.pow(opposite4.0, 2));59        double angleRad→ 0.9272952180016122 = Math.atan2(opposite4.0, adjacent3.0);60        double angleDeg→ 53.13010235415598 = Math.toDegrees(angleRad0.9272952180016122);61        62        System.out.println("Right triangle (3, 4, ?):");63        System.out.println("Hypotenuse: " + hypotenuse5.0);64        System.out.println("Angle: " + angleDeg53.13010235415598 + " degrees");6566        // Circle point67        double radius→ 10.0 = 10;68        double angleCircle→ 1.0471975511965976 = Math.toRadians(60);69        double x→ 5.000000000000001 = radius10.0 * Math.cos(angleCircle1.0471975511965976);70        double y→ 8.660254037844386 = radius10.0 * Math.sin(angleCircle1.0471975511965976);71        System.out.println("\nPoint on circle (r=10, 60°):");72        System.out.println("x: " + x5.000000000000001);73        System.out.println("y: " + y8.660254037844386);7475        // Distance and angle between points76        double x1 = 0, y1→ 0.0 = 0;77        double x2 = 3, y2→ 3.0 = 3;78        double distance→ 4.242640687119285 = Math.sqrt(Math.pow(x23.0 - x10.0, 2) + Math.pow(y23.0 - y10.0, 2));79        double angleToPoint→ 45.0 = Math.toDegrees(Math.atan2(y23.0 - y10.0, x23.0 - x10.0));80        81        System.out.println("\nFrom (0,0) to (3,3):");82        System.out.println("Distance: " + distance4.242640687119285);83        System.out.println("Angle: " + angleToPoint45.0 + " degrees");8485        // Table of sine values86        System.out.println("\nSine values (0° to 90°, step 15°):");87        for (int deg = 0; deg <= 90; deg += 15) {⋯outputAngle: 0.7853981633974483 radians (45 degrees)
Basic trigonometry:
sin(π/4): 0.7071067811865475
cos(π/4): 0.7071067811865476
tan(π/4): 0.9999999999999999

Common angles:
sin(0): 0.0
sin(π/2): 1.0
sin(π): 1.2246467991473532E-16
cos(0): 1.0
cos(π): -1.0

Inverse trigonometry:
asin(1): 1.5707963267948966
acos(0): 1.5707963267948966
atan(1): 0.7853981633974483
atan2(1, 1): 0.7853981633974483

Hyperbolic functions:
sinh(1): 1.1752011936438014
cosh(1): 1.543080634815244
tanh(1): 0.7615941559557649

Degree/radian conversion:
60.0 degrees = 1.0471975511965976 radians
1.0471975511965976 radians = 59.99999999999999 degrees

Trig with degrees (convert first):
sin(45°): 0.7071067811865475
cos(90°): 6.123233995736766E-17

Practical examples:
Right triangle (3, 4, ?):
Hypotenuse: 5.0
Angle: 53.13010235415598 degrees

Point on circle (r=10, 60°):
x: 5.000000000000001
y: 8.660254037844386

From (0,0) to (3,3):
Distance: 4.242640687119285
Angle: 45.0 degrees

Sine values (0° to 90°, step 15°):
2rad ← 0.0pass 1 of 7⋯86System.out.println("\nSine values (0° to 90°, step 15°):");87for (int deg0 = 0; deg <= 90; deg += 15) {88    double rad→ 0.0 = Math.toRadians(deg0);89    System.out.printf("%3d° : %.4f%n", deg0, Math.sin(rad0.0));90}⋯All 7 passes — pass 1 is the card abovepassdegrad
100.0
2150.2617993877991494
3300.5235987755982988
4450.7853981633974483
5601.0471975511965976
6751.3089969389957472
7901.5707963267948966

All 7 passes — pass 1 is the card above
pass	`deg`	`rad`
1	0	0.0
2	15	0.2617993877991494
3	30	0.5235987755982988
4	45	0.7853981633974483
5	60	1.0471975511965976
6	75	1.3089969389957472
7	90	1.5707963267948966

All 7 passes — pass 1 is the card above
pass	`deg`	`rad`
1	0	0.0
2	15	0.2617993877991494
3	30	0.5235987755982988
4	45	0.7853981633974483
5	60	1.0471975511965976
6	75	1.3089969389957472
7	90	1.5707963267948966

All 7 passes — pass 1 is the card above
pass	`deg`	`rad`
1	0	0.0
2	15	0.2617993877991494
3	30	0.5235987755982988
4	45	0.7853981633974483
5	60	1.0471975511965976
6	75	1.3089969389957472
7	90	1.5707963267948966

Radians The unit of angle measurement used by Math methods. Convert degrees to radians with Math.toRadians().

Special Functions

Additional mathematical operations for advanced calculations.

                            Special.java
                        
Replay: real traced execution (multi-file project)

                            // Special functions

public class Special {
    public static void main(String[] args) {
        // Hypotenuse
        System.out.println("Hypotenuse:");
        System.out.println("hypot(3, 4): " + Math.hypot(3, 4));
        System.out.println("hypot(5, 12): " + Math.hypot(5, 12));

        // Expm1 (exp(x) - 1) - accurate for small x
        System.out.println("\nExpm1 (e^x - 1):");
        System.out.println("expm1(0): " + Math.expm1(0));
        System.out.println("expm1(1): " + Math.expm1(1));
        System.out.println("expm1(0.001): " + Math.expm1(0.001));

        // Log1p (log(1 + x)) - accurate for small x
        System.out.println("\nLog1p (ln(1 + x)):");
        System.out.println("log1p(0): " + Math.log1p(0));
        System.out.println("log1p(1): " + Math.log1p(1));
        System.out.println("log1p(0.001): " + Math.log1p(0.001));

        // CopySign
        System.out.println("\nCopySign (magnitude of first, sign of second):");
        System.out.println("copySign(5, -1): " + Math.copySign(5, -1));    // -5.0
        System.out.println("copySign(-5, 1): " + Math.copySign(-5, 1));    // 5.0
        System.out.println("copySign(3.14, -2): " + Math.copySign(3.14, -2));

        // NextAfter
        System.out.println("\nNextAfter (next floating-point value):");
        double x = 1.0;
        System.out.println("nextAfter(1.0, 2.0): " + Math.nextAfter(x, 2.0));
        System.out.println("nextAfter(1.0, 0.0): " + Math.nextAfter(x, 0.0));

        // Ulp (unit in last place)
        System.out.println("\nUlp (spacing to next value):");
        System.out.println("ulp(1.0): " + Math.ulp(1.0));
        System.out.println("ulp(10.0): " + Math.ulp(10.0));
        System.out.println("ulp(100.0): " + Math.ulp(100.0));

        // Get exponent
        System.out.println("\nGet exponent:");
        System.out.println("getExponent(8.0): " + Math.getExponent(8.0));    // 3 (2^3 = 8)
        System.out.println("getExponent(16.0): " + Math.getExponent(16.0));  // 4
        System.out.println("getExponent(0.5): " + Math.getExponent(0.5));    // -1

        // Scalb (multiply by 2^n)
        System.out.println("\nScalb (multiply by 2^n):");
        System.out.println("scalb(3.0, 2): " + Math.scalb(3.0, 2));    // 3 * 2^2 = 12
        System.out.println("scalb(5.0, 3): " + Math.scalb(5.0, 3));    // 5 * 2^3 = 40
        System.out.println("scalb(1.0, -2): " + Math.scalb(1.0, -2));  // 1 * 2^-2 = 0.25

        // FMA (fused multiply-add) - a*b + c
        System.out.println("\nFMA (fused multiply-add):");
        System.out.println("fma(2, 3, 5): " + Math.fma(2, 3, 5));      // 2*3 + 5 = 11
        System.out.println("fma(1.5, 2, 0.5): " + Math.fma(1.5, 2, 0.5)); // 3.5

        // NextUp and NextDown
        System.out.println("\nNextUp and NextDown:");
        double val = 1.0;
        System.out.println("nextUp(1.0): " + Math.nextUp(val));
        System.out.println("nextDown(1.0): " + Math.nextDown(val));

        // Special values
        System.out.println("\nSpecial values:");
        System.out.println("Infinity: " + Double.POSITIVE_INFINITY);
        System.out.println("NaN: " + Double.NaN);
        System.out.println("Max double: " + Double.MAX_VALUE);
        System.out.println("Min double: " + Double.MIN_VALUE);

        // Check special values
        System.out.println("\nCheck special values:");
        System.out.println("isNaN(0/0): " + Double.isNaN(0.0 / 0.0));
        System.out.println("isInfinite(1/0): " + Double.isInfinite(1.0 / 0.0));
        System.out.println("isFinite(100): " + Double.isFinite(100.0));

        // Practical examples
        System.out.println("\nPractical examples:");

        // Calculate hypotenuse without overflow
        double a = 1e200;
        double b = 1e200;
        double hypot = Math.hypot(a, b);
        System.out.println("Hypotenuse of very large sides: " + hypot);

        // Accurate small calculations
        double small = 0.0001;
        double expResult = Math.expm1(small);
        double logResult = Math.log1p(small);
        System.out.println("expm1(" + small + "): " + expResult);
        System.out.println("log1p(" + small + "): " + logResult);

        // Apply sign
        double magnitude = 42;
        double sign = -1;
        double result = Math.copySign(magnitude, sign);
        System.out.println("Apply sign of " + sign + " to " + magnitude + ": " + result);
    }

    //help h1
    // Math.hypot(x, y) - sqrt(x^2 + y^2) without overflow
    // Math.expm1(x) - e^x - 1 (accurate for small x)
    // Math.log1p(x) - ln(1 + x) (accurate for small x)
    // Math.copySign(magnitude, sign)
    // Math.fma(a, b, c) - a*b + c (fused)
    // Math.scalb(x, n) - x * 2^n
    // Double.isNaN(), isInfinite(), isFinite()
    //end
}

                        
1x ← 1.0, val ← 1.0, a ← 1.0E200, b ← 1.0E200, hypot ← 1.414213562373095E200⋯3public class Special {4    public static void main(String[] args) {5        // Hypotenuse6        System.out.println("Hypotenuse:");7        System.out.println("hypot(3, 4): " + Math.hypot(3, 4));8        System.out.println("hypot(5, 12): " + Math.hypot(5, 12));910        // Expm1 (exp(x) - 1) - accurate for small x11        System.out.println("\nExpm1 (e^x - 1):");12        System.out.println("expm1(0): " + Math.expm1(0));13        System.out.println("expm1(1): " + Math.expm1(1));14        System.out.println("expm1(0.001): " + Math.expm1(0.001));1516        // Log1p (log(1 + x)) - accurate for small x17        System.out.println("\nLog1p (ln(1 + x)):");18        System.out.println("log1p(0): " + Math.log1p(0));19        System.out.println("log1p(1): " + Math.log1p(1));20        System.out.println("log1p(0.001): " + Math.log1p(0.001));2122        // CopySign23        System.out.println("\nCopySign (magnitude of first, sign of second):");24        System.out.println("copySign(5, -1): " + Math.copySign(5, -1));    // -5.025        System.out.println("copySign(-5, 1): " + Math.copySign(-5, 1));    // 5.026        System.out.println("copySign(3.14, -2): " + Math.copySign(3.14, -2));2728        // NextAfter29        System.out.println("\nNextAfter (next floating-point value):");30        double x→ 1.0 = 1.0;31        System.out.println("nextAfter(1.0, 2.0): " + Math.nextAfter(x1.0, 2.0));32        System.out.println("nextAfter(1.0, 0.0): " + Math.nextAfter(x1.0, 0.0));3334        // Ulp (unit in last place)35        System.out.println("\nUlp (spacing to next value):");36        System.out.println("ulp(1.0): " + Math.ulp(1.0));37        System.out.println("ulp(10.0): " + Math.ulp(10.0));38        System.out.println("ulp(100.0): " + Math.ulp(100.0));3940        // Get exponent41        System.out.println("\nGet exponent:");42        System.out.println("getExponent(8.0): " + Math.getExponent(8.0));    // 3 (2^3 = 8)43        System.out.println("getExponent(16.0): " + Math.getExponent(16.0));  // 444        System.out.println("getExponent(0.5): " + Math.getExponent(0.5));    // -14546        // Scalb (multiply by 2^n)47        System.out.println("\nScalb (multiply by 2^n):");48        System.out.println("scalb(3.0, 2): " + Math.scalb(3.0, 2));    // 3 * 2^2 = 1249        System.out.println("scalb(5.0, 3): " + Math.scalb(5.0, 3));    // 5 * 2^3 = 4050        System.out.println("scalb(1.0, -2): " + Math.scalb(1.0, -2));  // 1 * 2^-2 = 0.255152        // FMA (fused multiply-add) - a*b + c53        System.out.println("\nFMA (fused multiply-add):");54        System.out.println("fma(2, 3, 5): " + Math.fma(2, 3, 5));      // 2*3 + 5 = 1155        System.out.println("fma(1.5, 2, 0.5): " + Math.fma(1.5, 2, 0.5)); // 3.55657        // NextUp and NextDown58        System.out.println("\nNextUp and NextDown:");59        double val→ 1.0 = 1.0;60        System.out.println("nextUp(1.0): " + Math.nextUp(val1.0));61        System.out.println("nextDown(1.0): " + Math.nextDown(val1.0));6263        // Special values64        System.out.println("\nSpecial values:");65        System.out.println("Infinity: " + Double.POSITIVE_INFINITY);66        System.out.println("NaN: " + Double.NaN);67        System.out.println("Max double: " + Double.MAX_VALUE);68        System.out.println("Min double: " + Double.MIN_VALUE);6970        // Check special values71        System.out.println("\nCheck special values:");72        System.out.println("isNaN(0/0): " + Double.isNaN(0.0 / 0.0));73        System.out.println("isInfinite(1/0): " + Double.isInfinite(1.0 / 0.0));74        System.out.println("isFinite(100): " + Double.isFinite(100.0));7576        // Practical examples77        System.out.println("\nPractical examples:");78        79        // Calculate hypotenuse without overflow80        double a→ 1.0E200 = 1e200;81        double b→ 1.0E200 = 1e200;82        double hypot→ 1.414213562373095E200 = Math.hypot(a1.0E200, b1.0E200);83        System.out.println("Hypotenuse of very large sides: " + hypot1.414213562373095E200);8485        // Accurate small calculations86        double small→ 1.0E-4 = 0.0001;87        double expResult→ 1.0000500016667084E-4 = Math.expm1(small1.0E-4);88        double logResult→ 9.999500033330834E-5 = Math.log1p(small1.0E-4);89        System.out.println("expm1(" + small1.0E-4 + "): " + expResult1.0000500016667084E-4);90        System.out.println("log1p(" + small1.0E-4 + "): " + logResult9.999500033330834E-5);9192        // Apply sign93        double magnitude→ 42.0 = 42;94        double sign→ -1.0 = -1;95        double result→ -42.0 = Math.copySign(magnitude42.0, sign-1.0);96        System.out.println("Apply sign of " + sign-1.0 + " to " + magnitude42.0 + ": " + result-42.0);97    }⋯outputHypotenuse:
hypot(3, 4): 5.0
hypot(5, 12): 13.0

Expm1 (e^x - 1):
expm1(0): 0.0
expm1(1): 1.718281828459045
expm1(0.001): 0.0010005001667083417

Log1p (ln(1 + x)):
log1p(0): 0.0
log1p(1): 0.6931471805599453
log1p(0.001): 9.995003330835331E-4

CopySign (magnitude of first, sign of second):
copySign(5, -1): -5.0
copySign(-5, 1): 5.0
copySign(3.14, -2): -3.14

NextAfter (next floating-point value):
nextAfter(1.0, 2.0): 1.0000000000000002
nextAfter(1.0, 0.0): 0.9999999999999999

Ulp (spacing to next value):
ulp(1.0): 2.220446049250313E-16
ulp(10.0): 1.7763568394002505E-15
ulp(100.0): 1.4210854715202004E-14

Get exponent:
getExponent(8.0): 3
getExponent(16.0): 4
getExponent(0.5): -1

Scalb (multiply by 2^n):
scalb(3.0, 2): 12.0
scalb(5.0, 3): 40.0
scalb(1.0, -2): 0.25

FMA (fused multiply-add):
fma(2, 3, 5): 11.0
fma(1.5, 2, 0.5): 3.5

NextUp and NextDown:
nextUp(1.0): 1.0000000000000002
nextDown(1.0): 0.9999999999999999

Special values:
Infinity: Infinity
NaN: NaN
Max double: 1.7976931348623157E308
Min double: 4.9E-324

Check special values:
isNaN(0/0): true
isInfinite(1/0): true
isFinite(100): true

Practical examples:
Hypotenuse of very large sides: 1.414213562373095E200
expm1(1.0E-4): 1.0000500016667084E-4
log1p(1.0E-4): 9.999500033330834E-5
Apply sign of -1.0 to 42.0: -42.0

@seealso biginteger_intro, random_intro

Exercise: Practical.java

Calculate the distance between two points and the area of a circle