Introduction to Trigonometry

Radian Measure

$\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$

| Degrees | Radians | |---------|---------| | $0°$ | $0$ | | $30°$ | $\frac{\pi}{6}$ | | $45°$ | $\frac{\pi}{4}$ | | $60°$ | $\frac{\pi}{3}$ | | $90°$ | $\frac{\pi}{2}$ | | $180°$ | $\pi$ | | $360°$ | $2\pi$ |

The Unit Circle

A circle with radius 1 centered at the origin. For angle $\theta$: $\cos \theta = x\text{-coordinate} \quad \sin \theta = y\text{-coordinate}$

Key Values

| $\theta$ | $\sin \theta$ | $\cos \theta$ | $\tan \theta$ | |-----------|---------------|---------------|---------------| | $0$ | $0$ | $1$ | $0$ | | $\frac{\pi}{6}$ | $\frac{1}{2}$ | $\frac{\sqrt{3}}{2}$ | $\frac{\sqrt{3}}{3}$ | | $\frac{\pi}{4}$ | $\frac{\sqrt{2}}{2}$ | $\frac{\sqrt{2}}{2}$ | $1$ | | $\frac{\pi}{3}$ | $\frac{\sqrt{3}}{2}$ | $\frac{1}{2}$ | $\sqrt{3}$ | | $\frac{\pi}{2}$ | $1$ | $0$ | undefined |

Graphing Trig Functions

$y = A\sin(Bx - C) + D$

• Amplitude: $|A|$
• Period: $\frac{2\pi}{|B|}$
• Phase shift: $\frac{C}{B}$
• Vertical shift: $D$

Sine: Starts at 0, goes up

Cosine: Starts at max, goes down

Tangent: Has vertical asymptotes, period $\pi$

Identities to Know

Pythagorean Identity

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

Reciprocal Functions

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

Memory for signs: "All Students Take Calculus" — tells which trig functions are positive in quadrants I, II, III, IV.