Trigonometry Cheatsheet

Cheatsheet Content

1. Right Triangle Trigonometry (SOH CAH TOA) For a right triangle with angle $\theta$: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ Reciprocal Identities: $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Opposite}}$ $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Adjacent}}$ $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Adjacent}}{\text{Opposite}}$ Pythagorean Theorem: $a^2 + b^2 = c^2$ 2. Unit Circle Radius $r=1$ centered at origin $(0,0)$. Point $(x,y)$ on the circle at angle $\theta$: $x = \cos(\theta)$ $y = \sin(\theta)$ $\tan(\theta) = \frac{y}{x}$ Angles in radians: $2\pi \text{ rad} = 360^\circ$ Conversion: $\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ}$ Conversion: $\text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$ Common Angles $\theta$ (deg) $\theta$ (rad) $\sin(\theta)$ $\cos(\theta)$ $\tan(\theta)$ $0^\circ$ $0$ $0$ $1$ $0$ $30^\circ$ $\frac{\pi}{6}$ $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{\sqrt{3}}$ $45^\circ$ $\frac{\pi}{4}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ $60^\circ$ $\frac{\pi}{3}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $90^\circ$ $\frac{\pi}{2}$ $1$ $0$ Undefined 3. Trigonometric Identities Pythagorean Identities $\sin^2(\theta) + \cos^2(\theta) = 1$ $1 + \tan^2(\theta) = \sec^2(\theta)$ $1 + \cot^2(\theta) = \csc^2(\theta)$ Quotient Identities $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$ Even/Odd Identities $\sin(-\theta) = -\sin(\theta)$ (Odd) $\cos(-\theta) = \cos(\theta)$ (Even) $\tan(-\theta) = -\tan(\theta)$ (Odd) Cofunction Identities $\sin(\frac{\pi}{2} - \theta) = \cos(\theta)$ $\cos(\frac{\pi}{2} - \theta) = \sin(\theta)$ $\tan(\frac{\pi}{2} - \theta) = \cot(\theta)$ Sum and Difference Identities $\sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B)$ $\cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B)$ $\tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)}$ Double Angle Identities $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$ $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) = 2\cos^2(\theta) - 1 = 1 - 2\sin^2(\theta)$ $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$ Half Angle Identities $\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \implies \sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$ $\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \implies \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$ $\tan(\frac{\theta}{2}) = \frac{1 - \cos(\theta)}{\sin(\theta)} = \frac{\sin(\theta)}{1 + \cos(\theta)}$ 4. Inverse Trigonometric Functions $\arcsin(x)$ or $\sin^{-1}(x)$: domain $[-1, 1]$, range $[-\frac{\pi}{2}, \frac{\pi}{2}]$ $\arccos(x)$ or $\cos^{-1}(x)$: domain $[-1, 1]$, range $[0, \pi]$ $\arctan(x)$ or $\tan^{-1}(x)$: domain $(-\infty, \infty)$, range $(-\frac{\pi}{2}, \frac{\pi}{2})$ Properties $\sin(\arcsin(x)) = x$ for $x \in [-1, 1]$ $\arcsin(\sin(\theta)) = \theta$ for $\theta \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ $\arcsin(x) + \arccos(x) = \frac{\pi}{2}$ 5. Law of Sines and Cosines (General Triangles) Law of Sines $\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$ Law of Cosines $a^2 = b^2 + c^2 - 2bc\cos(A)$ $b^2 = a^2 + c^2 - 2ac\cos(B)$ $c^2 = a^2 + b^2 - 2ab\cos(C)$ 6. Area of a Triangle Given base $b$ and height $h$: $Area = \frac{1}{2}bh$ Given two sides and included angle: $Area = \frac{1}{2}ab\sin(C) = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B)$ Heron's Formula (given sides $a,b,c$): $s = \frac{a+b+c}{2}$ (semi-perimeter) $Area = \sqrt{s(s-a)(s-b)(s-c)}$ 7. Complex Numbers and Polar Coordinates Rectangular Form $z = x + iy$, where $i = \sqrt{-1}$ $x = \text{Re}(z)$, $y = \text{Im}(z)$ Polar Form $z = r(\cos(\theta) + i\sin(\theta))$ $r = |z| = \sqrt{x^2 + y^2}$ (magnitude/modulus) $\theta = \arg(z) = \arctan(\frac{y}{x})$ (argument/angle), adjust based on quadrant Conversion: $x = r\cos(\theta)$, $y = r\sin(\theta)$ Euler's Formula $e^{i\theta} = \cos(\theta) + i\sin(\theta)$ So, $z = re^{i\theta}$ De Moivre's Theorem $(r(\cos(\theta) + i\sin(\theta)))^n = r^n(\cos(n\theta) + i\sin(n\theta))$ Or, $(re^{i\theta})^n = r^ne^{in\theta}$ Used for finding powers and roots of complex numbers.