Sine & Cosine Waves

Drag the sliders to reshape the waves in real time

MAT 172 — Pre-Calc Trig
Every wave you'll ever graph is just y = A sin(Bx + C) + D or y = A cos(Bx + C) + D. Change one number, watch what happens. That's the whole chapter.

Sine wave
A 1
B 1
C 0
D 0
Cosine wave
A 1
B 1
C 0
D 0

Live wave properties

Sine amplitude
1
Cosine amplitude
1
Sine period
2π
Cosine period
2π
Sine phase shift
0
Cosine phase shift
0

The general form

y = A sin(Bx + C) + D
y = A cos(Bx + C) + D
A — amplitude (height). |A| = max distance from midline. Negative A flips the wave.
B — affects period. Period = 2π/B. Bigger B = faster wave.
C — phase shift = −C/B. Positive C shifts left, negative shifts right.
D — vertical shift. Moves the midline up or down.

Key vocabulary

Amplitude
|A| — the maximum distance from the midline to a peak or trough. Never negative.
Period
2π/B — the horizontal length of one complete cycle. Larger B = shorter period = more waves.
Phase shift
−C/B — how far the wave is shifted left or right from the origin.
Vertical shift
D — moves the entire wave up (positive D) or down (negative D). The midline is y = D.
Midline
The horizontal line y = D that the wave oscillates around. Not always the x-axis!

Things to try

Set B = 2 and watch the period cut in half. Set B = 0.5 and it doubles.
Set A = −1 to flip the sine wave upside down. The amplitude is still 1.
Sine and cosine are the same wave — just shifted. Set C = π/2 (≈1.57) on the sine and it looks like cosine.
Set D = 1 to shift the midline up to y = 1. Max becomes A+1, min becomes −A+1.
Exam tip: always find amplitude, period, phase shift, and vertical shift in that order before graphing.