The Graphs of Sine and Cosine

The height of a turning point

On the unit circle, the circle of radius 1 centered at the origin, a radius turned counterclockwise through an angle $\theta$θ ends at the point $\left(\cos \theta ,\sin \theta \right)$(cos θ, sin θ). So as the radius turns, the height of its end is always $\sin \theta$sin θ.

Follow that height through one full turn. At 0° the point is on the x-axis, at height 0. It rises to a height of 1 at 90°, the top of the circle. It comes back down to 0 at 180°, sinks to −1 at 270°, the bottom of the circle, and returns to 0 at 360°, where it started.

Unrolling the turn

Now plot that height against the angle. Put the angle $\theta$θ along the horizontal axis and the height $\sin \theta$sin θ up the vertical axis. Each angle gives one point: (0°, 0), (90°, 1), (180°, 0), (270°, −1) and (360°, 0) are the five key points.

Between them the heights change smoothly, because the point moves smoothly round the circle. Some values in between: sin 30° = ½ and sin 150° = ½, and sin 210° = −½ and sin 330° = −½. Joined up, the points make a smooth wave. This is the graph of $y=\sin \theta$y = sin θ.

The wave is steepest where it crosses the axis, at 0°, 180° and 360°, because there the point on the circle is moving straight up or straight down. It is flat at the top and the bottom, at 90° and 270°, because there the point is moving sideways and its height hardly changes. So the peaks are rounded, not pointed.

Left, the radius at 60°, with its height drawn in gold. Right, the same height plotted above 60° on the graph: sin 60° = 0.87, to 2 decimal places. Turn the radius and the graph is drawn behind it, up to 1 at 90° and down to −1 at 270°. The cos button plots the distance across instead. The tan button draws the graph of tangent.

$y=\sin \theta$y = sin θ from 0° to 360°. One square across is 90° and one square up is 1. The wave starts at 0, peaks at 1 at 90°, and reaches its lowest value, −1, at 270°.

Period and range

After a full turn of 360° the radius is back where it started, so everything repeats. The next turn draws exactly the same wave from 360° to 720°, and the one after that from 720° to 1080°. In symbols, $\sin \left(\theta +360^{\circ }\right)=\sin \theta$sin (θ + 360°) = sin θ for every angle $\theta$θ.

A graph that repeats itself is called periodic, and the length of one repeat is its period. The period of $y=\sin \theta$y = sin θ is 360°.

Turning the radius clockwise gives negative angles, and the wave continues to the left of the vertical axis in the same way: sin (−90°) = −1, the bottom of the circle.

The height of a point on a circle of radius 1 is never more than 1 or less than −1. So $\sin \theta$sin θ always lies between −1 and 1, and the range of $y=\sin \theta$y = sin θ is $-1\le y\le 1$−1 ≤ y ≤ 1. An equation such as $\sin \theta =2$sin θ = 2 has no solution.

Two full turns, from 0° to 720°, with one square across for every 90°. The second wave is an exact copy of the first, because a turn of 360° brings the point back to where it started.

The graph of cosine

Plot the distance across, $\cos \theta$cos θ, against the angle in the same way. At 0° the point is (1, 0), so the cosine starts at 1. It falls to 0 at 90°, reaches −1 at 180°, climbs back to 0 at 270° and is 1 again at 360°. The key points are (0°, 1), (90°, 0), (180°, −1), (270°, 0) and (360°, 1).

The graph of $y=\cos \theta$y = cos θ is the same wave as the graph of sine, with the same period of 360° and the same range from −1 to 1. It starts at its peak instead of on the axis.

In fact it is the sine wave slid 90° to the left. Each value the sine reaches at some angle, the cosine reached 90° earlier: the sine peaks at 90° and the cosine at 0°; the sine is at its lowest at 270° and the cosine at 180°.

The sine wave slid one square, 90°, to the left lands exactly on the cosine wave. One square across is 90°.

Why the shift is exactly 90°

Take the point at angle $\theta$θ, $\left(\cos \theta ,\sin \theta \right)$(cos θ, sin θ), and turn its radius a further 90° counterclockwise. A quarter turn about the origin takes a point (x, y) to (−y, x), so the new point is $\left(-\sin \theta ,\cos \theta \right)$(−sin θ, cos θ). But the new point is also the point at angle $\theta +90^{\circ }$θ + 90°, which is $\left(\cos \left(\theta +90^{\circ }\right),\sin \left(\theta +90^{\circ }\right)\right)$(cos (θ + 90°), sin (θ + 90°)).

Compare the heights: $\sin \left(\theta +90^{\circ }\right)=\cos \theta$sin (θ + 90°) = cos θ. The sine of an angle 90° larger equals the cosine, which is what "the cosine graph is the sine graph moved 90° to the left" means.

A check at one angle: cos 60° = ½, and sin (60° + 90°) = sin 150° = ½, the same.

Lines of symmetry

The sine graph is symmetrical about the vertical line through its peak at 90°. That is the fact $\sin \left(180^{\circ }-\theta \right)=\sin \theta$sin (180° − θ) = sin θ drawn on the graph: 30° and 150° are the same distance either side of 90°, and their sines are both ½.

The cosine graph is symmetrical about the vertical line through each of its peaks, at 0° and 360°, and through its lowest point at 180°. The line at 180° gives $\cos \left(360^{\circ }-\theta \right)=\cos \theta$cos (360° − θ) = cos θ: for example cos 60° and cos 300° are both ½. On the circle, the points at $\theta$θ and $360^{\circ }-\theta$360° − θ are reflections of each other in the x-axis, with the same distance across. These symmetries give the second angle with the same sine or cosine: $\sin \theta =\frac{1}{2}$sin θ = ½ at 30° and at 150°, and $\cos \theta =\frac{1}{2}$cos θ = ½ at 60° and at 300°.

Sketching the graphs

Mark the angle axis at 90°, 180°, 270° and 360°, and the vertical axis at 1 and −1. Plot the five key points: for sine 0, 1, 0, −1, 0, and for cosine 1, 0, −1, 0, 1. Join them with a smooth curve that is rounded at the top and the bottom and steepest where it crosses the axis.

The usual mistakes

Drawing straight lines between the key points, which gives pointed peaks. The wave is flat at its highest and lowest points.

Starting the cosine graph at 0. At 0° the point on the circle is (1, 0), so cos 0° = 1 and the cosine graph starts at its peak.

Letting the wave go above 1 or below −1. The height of a point on a circle of radius 1 is never more than 1.

Shifting the sine graph the wrong way. The cosine reaches each value 90° before the sine, so the cosine graph is the sine graph moved 90° to the left, not to the right.