Math Core

As a point travels around the unit circle, its height rises and falls, then repeats the same pattern on every lap. Graph that height against the angle and you get a wave. The sine and cosine graphs are the basic models for anything that repeats: tides, daylight hours, sound, alternating current.

Unwrapping the unit circle

Think of $x$ as the angle (in radians) and $y$ as the output. For $y = \sin x$ , the output is the height of the point on the unit circle at angle $x$ . Start at $x = 0$ and follow one lap:

From $0$ to $\dfrac{\pi}{2}$ , the point climbs from height $0$ to height $1$ .
From $\dfrac{\pi}{2}$ to $\pi$ , it falls back to $0$ .
From $\pi$ to $\dfrac{3\pi}{2}$ , it drops to $-1$ .
From $\dfrac{3\pi}{2}$ to $2\pi$ , it rises back to $0$ , where it started.

$x$	$0$	$\dfrac{\pi}{2}$	$\pi$	$\dfrac{3\pi}{2}$	$2\pi$
$\sin x$	$0$	$1$	$0$	$-1$	$0$
$\cos x$	$1$	$0$	$-1$	$0$	$1$

These five points (a start, a peak or valley, a crossing, and so on, every quarter lap) are the key points. Plot them, connect them with a smooth curve, and repeat the pattern in both directions.

y = sin x. It crosses the x-axis at multiples of π, peaks at π/2 and bottoms out at 3π/2.Open in grapher →

The cosine graph follows the $x$ -coordinate instead. It starts at its maximum, $\cos 0 = 1$ , falls to $-1$ at $x = \pi$ , and climbs back to $1$ at $x = 2\pi$ .

y = cos x. It starts at its maximum at x = 0 and crosses the x-axis at π/2 and 3π/2.Open in grapher →

Features of the graphs

Both curves are periodic: they repeat the same values forever. The length of one complete cycle is the period.

Key features of y = sin x and y = cos x

feature	$y = \sin x$	$y = \cos x$
domain	all real numbers	all real numbers
range	$-1 \le y \le 1$	$-1 \le y \le 1$
period	$2\pi$	$2\pi$
amplitude	$1$	$1$
midline	$y = 0$	$y = 0$
$x$ -intercepts	$x = k\pi$	$x = \dfrac{\pi}{2} + k\pi$
maximum points	$x = \dfrac{\pi}{2} + 2k\pi$	$x = 2k\pi$

Here $k$ is any integer.

The midline is the horizontal line halfway between the highest and lowest values. The amplitude is the distance from the midline to a maximum (or to a minimum). For the basic graphs, the midline is the $x$ -axis and the amplitude is $1$ .

Worked example: Reading values from the pattern

Use the pattern of $y = \sin x$ to find $\sin\dfrac{9\pi}{2}$ and $\sin(-\pi)$ .

Because the period is $2\pi$ , subtracting whole periods doesn't change the value:

\sin\frac{9\pi}{2} = \sin\left(\frac{9\pi}{2} - 4\pi\right) = \sin\frac{\pi}{2} = 1.

For $\sin(-\pi)$ : the sine graph crosses the $x$ -axis at every multiple of $\pi$ , so $\sin(-\pi) = 0$ .

Cosine is a shifted sine

Look at the two graphs together. The cosine graph is the sine graph slid $\dfrac{\pi}{2}$ units to the left. The peak of $y = \sin x$ at $x = \dfrac{\pi}{2}$ moves to $x = 0$ , which is the peak of $y = \cos x$ .

y = sin x and y = cos x. Each is a horizontal shift of the other by π/2.Open in grapher →

In symbols, $\cos x = \sin\left(x + \dfrac{\pi}{2}\right)$ for every $x$ . So in a sense there is only one wave shape, called a sinusoid, and sine and cosine are two different starting points on it.

Symmetry

The cosine graph is symmetric about the $y$ -axis, so $\cos(-x) = \cos x$ . Cosine is an even function. On the unit circle, the angles $x$ and $-x$ are reflections across the $x$ -axis, so they have the same $x$ -coordinate.

The sine graph is symmetric about the origin, so $\sin(-x) = -\sin x$ . Sine is an odd function. Reflecting across the $x$ -axis flips the sign of the $y$ -coordinate.

Worked example: Solving with the graph

Find all solutions of $\cos x = 0$ in the interval $0 \le x \le 2\pi$ , and describe where $\cos x$ is negative on that interval.

The cosine graph crosses the $x$ -axis at $x = \dfrac{\pi}{2}$ and $x = \dfrac{3\pi}{2}$ in this interval. Between those crossings the graph dips below the axis, so $\cos x < 0$ when $\dfrac{\pi}{2} < x < \dfrac{3\pi}{2}$ .

This matches the unit circle: those angles are in Quadrants II and III, where $x$ -coordinates are negative.

Common mistake

On the graph of $y = \sin x$ , the letter $x$ is the angle, not the $x$ -coordinate on the unit circle. The graph's horizontal axis measures angle in radians and its vertical axis shows the value of the function. Keep the two pictures separate: the circle shows one angle at a time, and the graph shows every angle at once.

Worked example: Counting solutions

How many solutions does $\sin x = \dfrac{1}{2}$ have in the interval $0 \le x \le 2\pi$ ? Find them.

Draw the horizontal line $y = \dfrac{1}{2}$ across one period of the sine graph. It crosses the rising part of the wave once and the falling part once, so there are two solutions. From the unit circle, the angles with height $\dfrac{1}{2}$ are $x = \dfrac{\pi}{6}$ (Quadrant I) and $x = \dfrac{5\pi}{6}$ (Quadrant II).

Tip

When you sketch either graph, mark the $x$ -axis in steps of $\dfrac{\pi}{2}$ . Every key point lands on one of those marks, so the sketch almost draws itself.