Trigonometric Ratios

Cheatsheet Content

1. Right Triangle Definitions For a right-angled triangle with an angle $\theta$: Hypotenuse (hyp): The side opposite the right angle. Opposite (opp): The side opposite to angle $\theta$. Adjacent (adj): The side next to angle $\theta$ (not the hypotenuse). Ratio Definition Mnemonic $\sin(\theta)$ $\frac{\text{opp}}{\text{hyp}}$ SOH $\cos(\theta)$ $\frac{\text{adj}}{\text{hyp}}$ CAH $\tan(\theta)$ $\frac{\text{opp}}{\text{adj}}$ TOA 2. Reciprocal Ratios These are the reciprocals of the primary trigonometric ratios: Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hyp}}{\text{opp}}$ Secant: $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hyp}}{\text{adj}}$ Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adj}}{\text{opp}}$ 3. Pythagorean Identities Fundamental identities derived from the Pythagorean theorem: $\sin^2(\theta) + \cos^2(\theta) = 1$ $1 + \tan^2(\theta) = \sec^2(\theta)$ $1 + \cot^2(\theta) = \csc^2(\theta)$ 4. Quotient Identities Expressing tangent and cotangent in terms of sine and cosine: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$ 5. Special Angles (in degrees and radians) $\theta$ $0^\circ \ (0)$ $30^\circ \ (\frac{\pi}{6})$ $45^\circ \ (\frac{\pi}{4})$ $60^\circ \ (\frac{\pi}{3})$ $90^\circ \ (\frac{\pi}{2})$ $\sin(\theta)$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$ $\cos(\theta)$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $\tan(\theta)$ $0$ $\frac{1}{\sqrt{3}}$ $1$ $\sqrt{3}$ Undefined 6. Unit Circle Definitions For a point $(x, y)$ on the unit circle ($r=1$) and angle $\theta$ from the positive x-axis: $\sin(\theta) = y$ $\cos(\theta) = x$ $\tan(\theta) = \frac{y}{x}$ For a circle with radius $r$: $\sin(\theta) = \frac{y}{r}$, $\cos(\theta) = \frac{x}{r}$, $\tan(\theta) = \frac{y}{x}$. 7. Angle Relationships 7.1. Cofunction Identities $\sin(90^\circ - \theta) = \cos(\theta)$ $\cos(90^\circ - \theta) = \sin(\theta)$ $\tan(90^\circ - \theta) = \cot(\theta)$ 7.2. Periodicity $\sin(\theta + 360^\circ k) = \sin(\theta)$ $\cos(\theta + 360^\circ k) = \cos(\theta)$ $\tan(\theta + 180^\circ k) = \tan(\theta)$ 7.3. Even/Odd Functions $\sin(-\theta) = -\sin(\theta)$ (Odd) $\cos(-\theta) = \cos(\theta)$ (Even) $\tan(-\theta) = -\tan(\theta)$ (Odd) 8. Law of Sines & Cosines (for any triangle) 8.1. Law of Sines $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ Where $a, b, c$ are side lengths and $A, B, C$ are opposite angles. 8.2. Law of Cosines $c^2 = a^2 + b^2 - 2ab \cos(C)$ $a^2 = b^2 + c^2 - 2bc \cos(A)$ $b^2 = a^2 + c^2 - 2ac \cos(B)$