Graphs of Sine and Cosine

Trace a point around the unit circle and record its height as the angle grows: you get the smooth, repeating wave of $y = \sin x$. Record its horizontal position instead and you get $y = \cos x$. These two graphs are the parent functions for every sinusoidal model in this course. All angles are in degrees.

Key ideas

Unwrapping the unit circle

For an angle $x$ in standard position, the point on the unit circle is $(\cos x, \sin x)$.

• As $x$ goes from $0^\circ$ to $90^\circ$, the point rises from height $0$ to $1$. From $90^\circ$ to $180^\circ$ it falls back to $0$, then down to $-1$ at $270^\circ$, and back to $0$ at $360^\circ$. Plotting the height against $x$ gives $y = \sin x$.
• The horizontal position starts at $1$, falls to $-1$ at $180^\circ$, and returns to $1$ at $360^\circ$. That’s $y = \cos x$.

After $360^\circ$, the point goes around again, so both graphs repeat.

The orange dots are the key points every $90^\circ$.

Key points

$x$	$0^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
$\sin x$	$0$	$1$	$0$	$-1$	$0$
$\cos x$	$1$	$0$	$-1$	$0$	$1$

Properties

Property	$y = \sin x$	$y = \cos x$
period	$360^\circ$	$360^\circ$
amplitude	$1$	$1$
axis	$y = 0$	$y = 0$
maximum	$1$, at $x = 90^\circ$	$1$, at $x = 0^\circ$ and $360^\circ$
minimum	$-1$, at $x = 270^\circ$	$-1$, at $x = 180^\circ$
zeros (from $0^\circ$ to $360^\circ$)	$0^\circ, 180^\circ, 360^\circ$	$90^\circ, 270^\circ$
domain	$\{x \in \mathbb{R}\}$	$\{x \in \mathbb{R}\}$
range	$\{y \in \mathbb{R} \mid -1 \le y \le 1\}$	$\{y \in \mathbb{R} \mid -1 \le y \le 1\}$

Sine and cosine are the same shape

The cosine graph is the sine graph shifted $90^\circ$ to the left:

$$\cos x = \sin(x + 90^\circ)$$

Worked examples

Example 1: A table of values

Make a table of $y = \sin x$ every $30^\circ$ from $0^\circ$ to $360^\circ$, to two decimal places.

Solution.

$x$	$0^\circ$	$30^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$150^\circ$	$180^\circ$
$\sin x$	$0$	$0.5$	$0.87$	$1$	$0.87$	$0.5$	$0$

$x$	$210^\circ$	$240^\circ$	$270^\circ$	$300^\circ$	$330^\circ$	$360^\circ$
$\sin x$	$-0.5$	$-0.87$	$-1$	$-0.87$	$-0.5$	$0$

The values are symmetric: the second half of the cycle is the first half with the signs flipped.

Example 2: Solving with the graph

Use the graph of $y = \sin x$ to solve $\sin x = 0.5$ for $0^\circ \le x \le 360^\circ$.

Solution. Draw the horizontal line $y = 0.5$. It crosses the sine curve twice in one cycle, at $x = 30^\circ$ and $x = 150^\circ$. (This matches finding angles from 0° to 360°.)

Example 3: Beyond one cycle

Find $\sin 390^\circ$, and list all the zeros of $y = \sin x$ for $-360^\circ \le x \le 360^\circ$.

Solution. The period is $360^\circ$, so $\sin 390^\circ = \sin(390^\circ - 360^\circ) = \sin 30^\circ = 0.5$.

The sine graph crosses the axis every $180^\circ$: $x = -360^\circ, -180^\circ, 0^\circ, 180^\circ, 360^\circ$.

Example 4: Where is cosine negative?

For $0^\circ \le x \le 360^\circ$, where is $\cos x$ negative?

Solution. From the graph, $y = \cos x$ is below the axis between its zeros at $90^\circ$ and $270^\circ$. So $\cos x \lt 0$ for $90^\circ \lt x \lt 270^\circ$. This matches CAST: cosine is negative in quadrants II and III.

Common mistakes

Starting the sine graph at $1$. $\sin 0^\circ = 0$, so the sine graph starts on the axis. It’s cosine that starts at its maximum.

Mixing up the zeros. Sine is zero at multiples of $180^\circ$; cosine is zero at $90^\circ$, $270^\circ$, and so on.

Drawing sharp corners. The graphs are smooth waves, rounded at the maximums and minimums.

Using radian mode. These graphs are in degrees. In radian mode, a calculator gives very different values.

Practice

1. (Warm-up) For $0^\circ \le x \le 360^\circ$, where does $y = \cos x$ reach its maximum, and what is it?

Solution

The maximum is $1$, at $x = 0^\circ$ and $x = 360^\circ$.

2. (Warm-up) List the zeros of $y = \sin x$ for $0^\circ \le x \le 360^\circ$.

Solution

$0^\circ$, $180^\circ$, and $360^\circ$.

3. (Warm-up) State the period and amplitude of $y = \sin x$.

Solution

Period $360^\circ$, amplitude $1$.

4. (Core) For $0^\circ \le x \le 360^\circ$, where is $\sin x$ negative?

Solution

$180^\circ \lt x \lt 360^\circ$ (quadrants III and IV).

5. (Core) Find $\cos 420^\circ$ and $\sin(-90^\circ)$.

Solution

$\cos 420^\circ = \cos 60^\circ = 0.5$. $\sin(-90^\circ) = \sin 270^\circ = -1$.

6. (Core) How many solutions does $\cos x = 0.3$ have for $0^\circ \le x \le 720^\circ$? Explain using the graph.

Solution

The line $y = 0.3$ crosses the cosine curve twice in each $360^\circ$ cycle, and $0^\circ$ to $720^\circ$ is two cycles. So there are $4$ solutions.

7. (Core) Check that $\cos x = \sin(x + 90^\circ)$ for $x = 0^\circ$, $90^\circ$, and $180^\circ$.

Solution

• $x = 0^\circ$: $\cos 0^\circ = 1$ and $\sin 90^\circ = 1$. ✓
• $x = 90^\circ$: $\cos 90^\circ = 0$ and $\sin 180^\circ = 0$. ✓
• $x = 180^\circ$: $\cos 180^\circ = -1$ and $\sin 270^\circ = -1$. ✓

8. (Challenge) Sketch $y = \sin x$ and $y = \cos x$ on the same axes for $0^\circ \le x \le 360^\circ$. Where do they cross?

Solution

They cross where $\sin x = \cos x$, which means $\tan x = 1$: at $x = 45^\circ$ (both $\tfrac{\sqrt{2}}{2}$) and $x = 225^\circ$ (both $-\tfrac{\sqrt{2}}{2}$).

9. (Challenge) Explain, using the unit circle, why $\sin(x + 360^\circ) = \sin x$ for every angle $x$.

Solution

Adding $360^\circ$ means rotating one extra full turn, which brings the terminal arm back to exactly the same position. The point on the unit circle is the same, so its height, $\sin x$, is the same. That’s why the period is $360^\circ$.