The unit circle and trigonometric graphs

Defining sine and cosine on the unit circle, signs in the four quadrants, related angles, and the graphs of y = sin x, y = cos x and y = tan x.

The unit circle

A point on the circle $x^2 + y^2 = 1$ at angle $\theta$ (measured anticlockwise from the positive $x$-axis) has coordinates $(\cos\theta, \sin\theta)$.

Example 1●●

Find exact values of $\sin 210^\circ$ and $\cos 300^\circ$.

$210^\circ$ is in quadrant 3 with related angle $30^\circ$: $\sin 210^\circ = -\tfrac{1}{2}$.

$300^\circ$ is in quadrant 4 with related angle $60^\circ$: $\cos 300^\circ = \tfrac{1}{2}$.

Graphs

y = sin x and y = cos x

Both repeat every 360° (the period) and range between −1 and 1 (amplitude 1). The cosine graph is the sine graph shifted 90° left.

Flashcards

Period of $sin x$

$360^\circ$

Period of $ an x$

$180^\circ$

Point on the unit circle

$(\cos\theta, \sin\theta)$

Practice

Exact value of $\sin 210^\circ$?
Exact value of $\cos 300^\circ$?
Period of $y = \sin x$ in degrees?
Maximum value of $y = \cos x$?
In which quadrant are both $\sin$ and $\cos$ negative?
$\sin 270^\circ$?
$\tan 225^\circ$?
Solve $\cos x = \tfrac{\sqrt{3}}{2}$ for $0^\circ \le x \le 360^\circ$.