Trigonometric Functions of Any Angle

By the end of this lesson, you will be able to define all six trig functions for any angle using the unit circle and coordinate definitions, evaluate trig functions for non-standard angles, and work with trig functions of negative angles.

Extension Beyond Acute Angles

The right-triangle definitions of trig functions only work for $0° < \theta < 90°$. The unit circle extends these definitions to all real numbers.

For any angle $\theta$ in standard position, with terminal side intersecting the unit circle at $(x, y)$:

$$\sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x} \quad (x \neq 0)$$

$$\csc\theta = \frac{1}{y}, \quad \sec\theta = \frac{1}{x}, \quad \cot\theta = \frac{x}{y}$$

For a point $(a, b)$ (not necessarily on the unit circle) at distance $r = \sqrt{a^2+b^2}$:

$$\sin\theta = \frac{b}{r}, \quad \cos\theta = \frac{a}{r}, \quad \tan\theta = \frac{b}{a}$$

Quadrant Analysis

QI	QII	QIII	QIV
All positive	$\sin > 0$	$\tan > 0$	$\cos > 0$

Signs follow ASTC.

Example 1 — Angle from a Point

The terminal side of $\theta$ passes through $(-3, 4)$. Find all six trig functions.

$r = \sqrt{9+16} = 5$

$\sin\theta = 4/5$, $\cos\theta = -3/5$, $\tan\theta = -4/3$

$\csc\theta = 5/4$, $\sec\theta = -5/3$, $\cot\theta = -3/4$

(QII: sin positive, cos and tan negative)

Example 2 — Angle from One Trig Value

$\tan\theta = -7/24$ and $\cos\theta < 0$. Find $\sin\theta$.

$\tan\theta < 0$ occurs in QII or QIV. Since $\cos\theta < 0$ as well, $\theta$ is in QII, where $\sin > 0$, $\cos < 0$, and therefore $\tan = \sin/\cos < 0$.

In QII the point $(x, y)$ on the terminal side has $x < 0$ and $y > 0$.

With $\tan = -7/24$: let $a = -24$ (adj, negative in QII), $b = 7$ (opp, positive).

$r = \sqrt{576 + 49} = 25$

$\sin\theta = 7/25$

Example 3 — Functions at Quadrantal Angles

Angle	$\cos$