Trigonometric functions of a general angle

Define sin, cos and tan for any angle using a point on a circle, find values at the boundary angles 0°, 90°, 180° and 270°, and meet the graphs, domains, periods and amplitudes.

Angles on the coordinate plane

Measure $\theta$ anticlockwise from the positive $x$-axis. Negative angles go clockwise. Let the ray at angle $\theta$ meet the circle $x^2 + y^2 = r^2$ at $P(x, y)$. Then:

General definition

$$\cos\theta = \frac{x}{r} \qquad \sin\theta = \frac{y}{r} \qquad \tan\theta = \frac{y}{x}$$

On the unit circle ($r = 1$) this is simply $P = (\cos\theta, \sin\theta)$. For acute angles, these agree with SOH CAH TOA.

Example 1●

The point $P(-8, -15)$ lies on the ray at angle $\theta$. Find $\sin\theta$, $\cos\theta$ and $\tan\theta$.

$r = \sqrt{(-8)^2 + (-15)^2} = \sqrt{289} = 17$

r is the distance from the origin to P, so use Pythagoras.

$\sin\theta = \dfrac{y}{r} = -\dfrac{15}{17}$, $\quad \cos\theta = \dfrac{x}{r} = -\dfrac{8}{17}$

Use the general definition with x = −8, y = −15 and r = 17.

$\tan\theta = \dfrac{y}{x} = \dfrac{-15}{-8} = \dfrac{15}{8}$

Two negatives divide to give a positive.

The unit circle

The blue ray is at angle θ. Its endpoint P on the unit circle has coordinates (cos θ, sin θ).

Try this

Find the four angles where one coordinate of $P$ is $0$. These are the boundary angles.
In which quadrants is $\cos\theta$ negative?

Boundary angles

$\theta$	$0^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
$\sin\theta$	$0$	$1$	$0$	$-1$	$0$
$\cos\theta$	$1$	$0$	$-1$	$0$	$1$
$\tan\theta$	$0$	undefined	$0$	undefined	$0$

Example 2●●

Find the exact value of $3\sin 270^\circ - 2\cos 180^\circ + \tan 360^\circ$.

$\sin 270^\circ = -1$, $\quad \cos 180^\circ = -1$, $\quad \tan 360^\circ = 0$

Read each value from the boundary-angle table, or picture P on the unit circle.

$3(-1) - 2(-1) + 0 = -3 + 2 = -1$

Substitute, taking care with the double negative.

Graphs, period and amplitude

y = sin θ, y = cos θ and y = tan θ

Orange: sin. Blue: cos. Faint green: tan, with asymptotes at 90° and 270° (and every 180° from them).

$\sin\theta$ and $\cos\theta$ are defined for all angles, with range $-1 \le y \le 1$. Their period is $360^\circ$ and their amplitude is $1$.
$\tan\theta$ is undefined when $\cos\theta = 0$ (at $90^\circ$, $270^\circ$, …). Its period is $180^\circ$ and it has no amplitude.
$\sin$ and $\tan$ are odd functions; $\cos$ is even.

Flashcards

Point on the unit circle at angle θ

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

tan θ in terms of x and y

$\dfrac{y}{x}$

Period of sin and cos

$360^\circ$

Period of tan

$180^\circ$

cos 180°

$-1$

Practice

Find $\sin 270^\circ$.
Find $\cos 90^\circ$.
$\tan 90^\circ$ is:
The point $(-3, 4)$ lies on a ray at angle $\theta$. Then $\cos\theta =$
The amplitude of $y = \cos\theta$ is:
The period of $y = \tan\theta$ is:
With $P = (-3, 4)$ as above, find $\tan\theta$.
Which function is even?