Math Core

The unit circle gives you sine and cosine one angle at a time. If you plot those values against the angle, you get a smooth wave that repeats forever. That wave is the shape behind sound, tides, seasons and anything else that comes back around.

From the unit circle to a wave

Remember that the point on the unit circle at angle $x$ (in radians) is $(\cos x, \sin x)$ . So $\sin x$ is the height of that point, and $\cos x$ is its horizontal position.

Walk once around the circle, starting at $(1, 0)$ , and watch the height:

At $x = 0$ the point is at $(1, 0)$ , so the height is $0$ .
It climbs to the top of the circle, height $1$ , at $x = \dfrac{\pi}{2}$ .
It comes back down to height $0$ at $x = \pi$ .
It drops to the bottom, height $-1$ , at $x = \dfrac{3\pi}{2}$ .
It returns to height $0$ at $x = 2\pi$ , back where it started.

Now put the angle on the horizontal axis and the height on the vertical axis. The heights trace out the graph of $y = \sin x$ . Here are a few more values from the unit circle:

$x$	$0$	$\frac{\pi}{6}$	$\frac{\pi}{2}$	$\frac{5\pi}{6}$	$\pi$	$\frac{7\pi}{6}$	$\frac{3\pi}{2}$	$\frac{11\pi}{6}$	$2\pi$
$\sin x$	$0$	$0.5$	$1$	$0.5$	$0$	$-0.5$	$-1$	$-0.5$	$0$
$\cos x$	$1$	$\approx 0.87$	$0$	$\approx -0.87$	$-1$	$\approx -0.87$	$0$	$\approx 0.87$	$1$

Doing the same with the horizontal position gives $y = \cos x$ .

y = sin x (starts at 0 and rises) and y = cos x (starts at its maximum, 1).Open in grapher →

Five key points per cycle

You never need a long table to sketch these graphs. One full cycle is pinned down by five points, spaced a quarter of a turn apart: a zero, a peak, a zero, a valley, a zero (for sine), or a peak, a zero, a valley, a zero, a peak (for cosine).

Key points on [0, 2π]

$x$	$0$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$
$y = \sin x$	$0$	$1$	$0$	$-1$	$0$
$y = \cos x$	$1$	$0$	$-1$	$0$	$1$

Connect them with a smooth, rounded curve (never with sharp corners), then repeat the pattern to the left and right.

A handy phrase: sine starts in the middle and goes up; cosine starts at the top.

Periodic functions

After $x = 2\pi$ the point on the unit circle starts its second lap and every value repeats. Graphs with this property have a special name.

Definition

Periodic function

A function $f$ is periodic if there is a positive number $p$ with $f(x + p) = f(x)$ for every $x$ in its domain. The smallest such $p$ is the period. One copy of the repeating piece is called a cycle.

Both sine and cosine have period $2\pi$ :

\sin(x + 2\pi) = \sin x \qquad \cos(x + 2\pi) = \cos x

This lets you evaluate at large angles. Subtract (or add) multiples of $2\pi$ until you land in $[0, 2\pi)$ , then read the value from the key points.

Properties side by side

	$y = \sin x$	$y = \cos x$
Domain	all real numbers	all real numbers
Range	$[-1, 1]$	$[-1, 1]$
Period	$2\pi$	$2\pi$
Zeros	$x = k\pi$	$x = \frac{\pi}{2} + k\pi$
Maximum $1$ at	$x = \frac{\pi}{2} + 2k\pi$	$x = 2k\pi$
Minimum $-1$ at	$x = \frac{3\pi}{2} + 2k\pi$	$x = \pi + 2k\pi$
Symmetry	odd (origin)	even ( $y$ -axis)

Here $k$ stands for any integer. The midline of both graphs is the $x$ -axis, $y = 0$ : the curve spends equal time above and below it.

Even and odd

The cosine graph is a mirror image across the $y$ -axis, so $\cos(-x) = \cos x$ . The sine graph has rotational symmetry about the origin, so $\sin(-x) = -\sin x$ . On the unit circle, going clockwise instead of counterclockwise flips the height but keeps the horizontal position.

Cosine is a shifted sine

Look at the graph again: the cosine curve is exactly the sine curve slid $\dfrac{\pi}{2}$ units to the left. In symbols, $\cos x = \sin\left(x + \dfrac{\pi}{2}\right)$ . The two graphs have the same shape; they just start at different points in the cycle.

Worked example: Using periodicity

Find $\sin\left(\dfrac{7\pi}{2}\right)$ and $\cos(-3\pi)$ .

Solution. Since $\dfrac{7\pi}{2} = \dfrac{3\pi}{2} + 2\pi$ , the value matches $\sin\left(\dfrac{3\pi}{2}\right) = -1$ .

Cosine is even, so $\cos(-3\pi) = \cos(3\pi)$ . Subtract $2\pi$ : $\cos(3\pi) = \cos(\pi) = -1$ .

Worked example: Finding zeros on an interval

Find every $x$ in $[-2\pi, 2\pi]$ with $\cos x = 0$ .

Solution. The zeros of cosine are at $\dfrac{\pi}{2} + k\pi$ . Try integers $k$ and keep the values inside the interval:

$k = 0$ : $\dfrac{\pi}{2}$ . $k = 1$ : $\dfrac{3\pi}{2}$ . $k = 2$ : $\dfrac{5\pi}{2}$ , too big.
$k = -1$ : $-\dfrac{\pi}{2}$ . $k = -2$ : $-\dfrac{3\pi}{2}$ . $k = -3$ : $-\dfrac{5\pi}{2}$ , too small.

The zeros are $x = -\dfrac{3\pi}{2}, -\dfrac{\pi}{2}, \dfrac{\pi}{2}, \dfrac{3\pi}{2}$ .

Worked example: Where the graph rises and falls

On $[0, 2\pi]$ , where is $y = \sin x$ increasing?

Solution. Follow the key points. The graph rises from $(0, 0)$ to the peak at $\left(\dfrac{\pi}{2}, 1\right)$ , falls through $(\pi, 0)$ to the valley at $\left(\dfrac{3\pi}{2}, -1\right)$ , then rises again to $(2\pi, 0)$ . So sine is increasing on $\left[0, \dfrac{\pi}{2}\right]$ and on $\left[\dfrac{3\pi}{2}, 2\pi\right]$ , and decreasing on $\left[\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right]$ .

Worked example: Counting intersections

How many solutions does $\sin x = 0.4$ have on $[0, 4\pi]$ ?

Solution. Picture the horizontal line $y = 0.4$ . In one cycle the sine curve goes up past $0.4$ once (on the way to the peak) and back down past $0.4$ once (on the way down). That is $2$ crossings per cycle. The interval $[0, 4\pi]$ holds exactly $2$ cycles, so there are $2 \cdot 2 = 4$ solutions.

Common mistake

The horizontal axis is measured in radians, not degrees. One cycle ends at $2\pi \approx 6.28$ , and the first peak of sine is at $\dfrac{\pi}{2} \approx 1.57$ . When you sketch on graph paper, mark $\pi \approx 3.14$ carefully instead of placing $\pi$ at $x = 3$ or $x = 180$ .

Tip

To check a sketch, test one point. For example, $\cos\left(\dfrac{\pi}{3}\right) = 0.5$ , so your cosine curve should pass through about $(1.05, 0.5)$ , halfway down from the peak.

Practice

Practice 1

What is the range of $y = \cos x$ ?

Practice 2

Use the key points of the sine graph to find $\sin\left(\dfrac{3\pi}{2}\right)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

Find every zero of $y = \cos x$ in the interval $[0, 2\pi]$ . Enter your answers separated by commas (you can type pi).

Separate answers with commas, e.g. 2, -5

Practice 4

On $[0, 2\pi]$ , at what value of $x$ does $y = \cos x$ reach its minimum? (You can type pi.)

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Find $\sin\left(\dfrac{13\pi}{2}\right)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 6

Which statement is true for every real number $x$ ?

Practice 7

On $[0, 2\pi]$ , on which interval is $y = \cos x$ increasing?

Practice 8

How many solutions does $\cos x = -0.3$ have on the interval $[0, 6\pi]$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.