The Six Trigonometric Functions

By the end of this lesson, you will be able to define all six trig functions (sin, cos, tan, csc, sec, cot), compute their values for any quadrant angle, identify when each is undefined, and apply the Pythagorean identities.

The Six Functions

From the unit circle point $(\cos\theta, \sin\theta)$ and $\tan\theta = \sin\theta/\cos\theta$, we get all six:

Function	Definition	Reciprocal
$\sin\theta$	$y$-coordinate on unit circle	$\csc\theta = 1/\sin\theta$
$\cos\theta$	$x$-coordinate on unit circle	$\sec\theta = 1/\cos\theta$
$\tan\theta$	$\sin\theta/\cos\theta$	$\cot\theta = \cos\theta/\sin\theta$

Right triangle form: (for acute angles)

$\sin = \dfrac{\text{opp}}{\text{hyp}}$, $\cos = \dfrac{\text{adj}}{\text{hyp}}$, $\tan = \dfrac{\text{opp}}{\text{adj}}$

$\csc = \dfrac{\text{hyp}}{\text{opp}}$, $\sec = \dfrac{\text{hyp}}{\text{adj}}$, $\cot = \dfrac{\text{adj}}{\text{opp}}$

Undefined Values

$\tan\theta$ and $\sec\theta$ are undefined when $\cos\theta = 0$ (i.e., $\theta = \pi/2, 3\pi/2$, ...).

$\cot\theta$ and $\csc\theta$ are undefined when $\sin\theta = 0$ (i.e., $\theta = 0, \pi$, ...).

The Fundamental Pythagorean Identity

From $x^2 + y^2 = 1$ (unit circle) with $x = \cos\theta$, $y = \sin\theta$:

$$\sin^2\theta + \cos^2\theta = 1$$

Divide by $\cos^2\theta$: $\tan^2\theta + 1 = \sec^2\theta$

Divide by $\sin^2\theta$: $1 + \cot^2\theta = \csc^2\theta$

Example 1 — Finding All Six from One Value

$\sin\theta = 3/5$ and $\theta$ is in QI. Find all six trig functions.

$\cos\theta = 4/5$ (Pythagorean theorem or identity)

$\tan\theta = 3/4$

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

Example 2 — QII Angle

$\cos\theta = -12/13$, $\theta$ in QII. Find $\sin\theta$ and $\tan\theta$.

$\sin^2\theta = 1 - 144/169 = 25/169 \implies \sin\theta = 5/13$ (positive in QII)

$\tan\theta = \frac{5/13}{-12/13} = -5/12$ (negative in QII)

Example 3 — Exact Values from Unit Circle

Find all six trig functions for $\theta = 4\pi/3$.

$4\pi/3 = 240°$, in QIII. Reference angle: $60°$.

In QIII: sin and cos are negative; tan is positive.

$\sin(4\pi/3) = -\frac{\sqrt{3}}{2}$, $\cos(4\pi/3) = -\frac{1}{2}$, $\tan(4\pi/3) = \sqrt{3}$