Trigonometry — Unit Circle & Key Angles

The Unit Circle

A circle with radius $r = 1$ centered at the origin. For any angle $\theta$, the point on the circle is:

$$\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix}$$

$$60°$$ $$(\cos 60°, \sin 60°)$$ $$y = \sin 60°$$ $$x = \cos 60°$$ $$x\;(\cos\theta)$$ $$y\;(\sin\theta)$$

The fundamental identity follows from the Pythagorean theorem:

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

Key Angle Values

$\theta$ (degrees)	$\theta$ (radians)	$\sin\theta$	$\cos\theta$	$\tan\theta$
0°	$0$	$0$	$1$	$0$
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
90°	$\frac{\pi}{2}$	$1$	$0$	undefined

Sine and Cosine Waves

$$f(\theta) = \sin\theta \quad \text{and} \quad g(\theta) = \cos\theta$$

$$\frac{\pi}{6}$$ $$\frac{\pi}{3}$$ $$\frac{\pi}{2}$$ $$\frac{2\pi}{3}$$ $$\pi$$ $$\frac{4\pi}{3}$$ $$\frac{3\pi}{2}$$ $$\frac{5\pi}{3}$$ $$\theta \text{ (radians)}$$ $$\sin\theta$$ $$\cos\theta$$

The sine and cosine functions are periodic with period $2\pi$:

$$\sin(\theta + 2\pi) = \sin\theta, \quad \cos(\theta + 2\pi) = \cos\theta$$

Right Triangle Definitions

In a right triangle with angle $\theta$:

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$

$$\theta$$ $$\text{adjacent}$$ $$\text{opposite}$$

The tangent function relates sine and cosine:

$$\tan\theta = \frac{\sin\theta}{\cos\theta}$$

Important Identities

Pythagorean Identity:

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

Reciprocal Identities:

$$\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$$

Angle Sum Formulas:

$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$ $$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$