Sine and cosine on the unit circle

The unit circle

Why does a point going round a circle draw a sine wave?

Turn the angle θ and follow the height of a point on the unit circle. Unrolled along the angle, that height is the sine wave; the horizontal shadow is cosine.

Drag the ringed points on the board.

Middle school
High school
Unit circle
Radian measure
Special angles
sin² + cos² = 1

\theta

sin θ ≥ 0

sin θ ≤ 0

0.90

P = (\cos\theta,\ \sin\theta)

$\theta\ (\text{rad})$: 0.900
$\theta\ (^\circ)$: 51.6
$\sin\theta$: 0.783
$\cos\theta$: 0.622

Guided tour

1Thirty degrees
At $\theta=\frac{\pi}{6}$ (30°) the point $P$ sits at height $\frac12$ . The red segment is $\sin\theta$ ; the blue one, $\cos\theta=\frac{\sqrt3}{2}\approx0.866$ .
2The top
At $\theta=\frac{\pi}{2}$ the point is straight up: $\sin\theta=1$ , $\cos\theta=0$ . On the right, the wave reaches its peak at the same moment.
3Half a turn
At $\theta=\pi$ the point is on the left and its height is back to $0$ . The wave crosses the axis: half a period done.
4Below the axis
At $\theta=\frac{3\pi}{2}$ the point is at the bottom, so $\sin\theta=-1$ . After $2\pi$ the point is home and the wave starts over: the period is $2\pi$ .

Explanation

1
Sine and cosine as coordinates
Draw a circle of radius $1$ around the origin. Turn a ray by the angle $\theta$ from the positive $x$ -axis; it meets the circle at one point $P$ . By definition
$P=(\cos\theta,\ \sin\theta)$
This works for every angle, not just the acute angles of a right triangle, which is why the unit circle is the definition used beyond the first quadrant.
2
Unrolling the height
Plot the height of $P$ against the angle turned so far. Each full turn repeats the same heights, so the graph repeats every $2\pi$ : that is the sine wave. Its highest value is $1$ and its lowest is $-1$ , the radius of the circle.
The angle is measured in radians: an angle of $\theta$ cuts an arc of length $\theta$ on the unit circle, so the horizontal axis of the wave is literally the distance travelled around the circle.
3
Special angles worth knowing
$\begin{array}{c|ccccc}\theta & 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2}\\\hline \sin\theta & 0 & \frac12 & \frac{\sqrt2}{2} & \frac{\sqrt3}{2} & 1\\ \cos\theta & 1 & \frac{\sqrt3}{2} & \frac{\sqrt2}{2} & \frac12 & 0\end{array}$
Sine climbs while cosine falls by the same table read backwards, because $\cos\theta=\sin\left(\frac{\pi}{2}-\theta\right)$ .
4
The Pythagorean identity
The blue segment, the red segment and the radius form a right triangle with hypotenuse $1$ . Pythagoras gives, for every angle,
$\sin^2\theta+\cos^2\theta=1$

Check yourself

Q01

What is $\sin\frac{5\pi}{6}$ ?

Q02

What is the period of $y=\sin x$ ?

Q03

If $\sin\theta=0.6$ and $\theta$ is acute, what is $\cos\theta$ ?

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