Trig Functions

Sine and Cosine Graphs

Graph y = A sin(x): identify amplitude and period; find the maximum and minimum points over one complete period [0, 2π].

sine-cosine-graphs For y = A sin(x): amplitude = A (height from midline to peak), period = 2π (length of one complete cycle). Key points: max at x = π/2, min at x = 3π/2.

Example

Describe sine graph features as exact derived-state values. The amplitude is the absolute value of A, so a negative leading coefficient still gives a positive amplitude.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

y=2sin⁡xy= 2 \sin x

Step 2 — Amplitude

The amplitude is |2| = 2.

∣A∣=∣2∣=2|A|=| 2 |= \hl{2}

Step 3 — Period

Sine completes one cycle every 2 pi.

period=2π\text{period}= \hl{2} \hlmath{\pi}

Step 4 — Key points

Record the maximum and minimum points.

xyroleπ22maximum3π2−2minimum\hlmath{\begin{array}{c|c|c}x&y&\text{role}\\\frac{\pi}{2}&2&\text{maximum}\\\frac{3\pi}{2}&-2&\text{minimum}\end{array}}