Pythagoras
$$c^2 = a^2 + b^2$$
Trigonometric ratios (right-angled triangle)
$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan\theta = \frac{\sin\theta}{\cos\theta} \qquad \sin^2\theta + \cos^2\theta = 1 \qquad \sin\theta = \cos(90^\circ - \theta)$$
Special angles
| $\theta$ | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ |
|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $\tfrac{1}{2}$ | $\tfrac{\sqrt{2}}{2}$ | $\tfrac{\sqrt{3}}{2}$ | $1$ |
| $\cos\theta$ | $1$ | $\tfrac{\sqrt{3}}{2}$ | $\tfrac{\sqrt{2}}{2}$ | $\tfrac{1}{2}$ | $0$ |
| $\tan\theta$ | $0$ | $\tfrac{\sqrt{3}}{3}$ | $1$ | $\sqrt{3}$ | undefined |
Any triangle (sides $a, b, c$ opposite angles $A, B, C$)
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \quad\text{(sine rule)}$$ $$a^2 = b^2 + c^2 - 2bc\cos A \quad\text{(cosine rule)} \qquad \cos A = \frac{b^2 + c^2 - a^2}{2bc}$$ $$\text{Area} = \tfrac{1}{2}ab\sin C$$
Double angle and graphs
$$\sin 2\theta = 2\sin\theta\cos\theta \qquad \cos 2\theta = \cos^2\theta - \sin^2\theta = 1 - 2\sin^2\theta = 2\cos^2\theta - 1$$ For $y = a\sin(bx + c)$ (or cosine): amplitude $= |a|$, period $= \dfrac{360^\circ}{b}$ (or $\dfrac{2\pi}{b}$). For $y = a\tan bx$: period $= \dfrac{180^\circ}{b}$.
Angles of elevation and depression
The angle of elevation equals the angle of depression (alternate angles); use $\tan\theta = \dfrac{\text{height}}{\text{horizontal distance}}$.