Trig Equations

Solve Basic Trig Equation

Solve sin(x) = c on [0, 2π) by finding the reference angle and all solutions in the interval — both QI and QII for sin > 0.

solve-basic-trig-equation To solve sin(x) = c on [0, 2π): 1. Find the reference angle α with sin(α) = c (from table). 2. For sin > 0: solutions in QI (x = α) and QII (x = π - α). 3. Verify both solutions satisfy the original equation.

Example

Find every exact solution in the requested interval.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

sin⁡x=12x∈[0,2π)\sin x= \frac{1}{2} \quad x\in[ 0 , 2 \pi )

Step 2 — Reference angle

Use the reference angle: sine of pi over 6 is 1 over 2.

sin⁡(π6)=12\sin( \hlmath{\frac{\pi}{6}} )= \frac{1}{2}

Step 3 — Solution set

List both solutions in the interval.

x∈{π6,5π6}x\in\{ \hlmath{\frac{\pi}{6}} , \hlmath{\frac{5\pi}{6}} \}

Step 4 — Verify

Check pi over 6 and 5 pi over 6 both work.

sin⁡(π6)=sin⁡(5π6)=12\sin( \hlmath{\frac{\pi}{6}} )=\sin( \hlmath{\frac{5\pi}{6}} )= \frac{1}{2}