Graphs of Sine and Cosine: Amplitude, Period and Shifts

Q: What is the period of sin x?

2π (or 360°). The graph repeats after one full turn round the unit circle.

Q: How do I find the amplitude of a sine graph?

It is |A| in y = A sin (Bx), or half the distance from the lowest point to the highest point on the graph.

How unrolling the unit circle draws the sine wave, how cosine is the same wave shifted, and how amplitude, period and shifts change the graph, with an explorer to try each one.

3 min read Supports 3 lessons

Imagine the point on the unit circle travelling round and round. Plot its height against the angle it has turned through, and you draw a smooth wave: the graph of sine. Plot its horizontal position instead and you draw cosine.

Both waves repeat every $2\pi$ and stay between $-1$ and $1$ .

Key features

Sine starts at $0$ , rises to $1$ at $\frac{\pi}{2}$ , returns to $0$ at $\pi$ , falls to $-1$ at $\frac{3\pi}{2}$ , and is back at $0$ after one full turn, $2\pi$ . Then it repeats for ever. A graph that repeats like this is periodic.

Key idea

Cosine is the same wave as sine, shifted left by $\frac{\pi}{2}$ : $\cos x = \sin\!\left(x + \frac{\pi}{2}\right)$ .

Amplitude and period

Sinusoidal parameters

y = A\sin\big(B(x - h)\big) + k \qquad \text{amplitude } |A|,\ \text{period } \frac{2\pi}{|B|}

The amplitude is how far the wave rises above its centre line. The period is how long it takes to repeat. Bigger $B$ squeezes more waves into the same space, so the period gets shorter.

Shifts

$h$ slides the wave sideways (a phase shift) and $k$ lifts it, moving the centre line to $y = k$ . These are the same transformations as for any graph. Try them here:

$y = 3\sin\big(2(x - \tfrac{\pi}{4})\big) + 1$
Read off each feature.

1/4

Where waves appear

Tides, sound, alternating current, the seasons and the motion of a spring all follow sine waves. Adjusting $A$ , $B$ , $h$ and $k$ fits the model to the data: for daylight hours, the amplitude is the seasonal swing and the period is a year.

Watch out

The period depends on $B$ , not $A$ . Doubling $A$ makes the wave taller; it does not make it longer.

Common questions

What is the period of sin x?: $2\pi$ (or $360°$ ). The graph repeats after one full turn round the unit circle.
How do I find the amplitude of a sine graph?: It is $|A|$ in $y = A\sin(Bx)$ , or half the distance from the lowest point to the highest point on the graph.

Practise sine and cosine graphs

The Graphs of Sine & Cosine lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

What is the period of sin x?

$2\pi$ (or $360°$ ). The graph repeats after one full turn round the unit circle.

How do I find the amplitude of a sine graph?

It is $|A|$ in $y = A\sin(Bx)$ , or half the distance from the lowest point to the highest point on the graph.