Trigonometry

This section contains a reference for trigonometric identities.

Triangle Identities

With reference to figure A.1, the law of sines is $$\begin{align} \frac{\sin\alpha} {a} = \frac{\sin\beta} {b} = \frac{\sin\gamma} {c} \end{align}$$ and the law of cosines is $$\begin{align} c^2 &= a^2 + b^2 - 2 a b \ \cos\gamma \\ b^2 &= a^2 + c^2 - 2 a c \ \cos\beta \\ a^2 &= c^2 + b^2 - 2 c b \ \cos\alpha \end{align}$$

Reciprocal Identities

$$\begin{align} \csc u &= \frac{1} {\sin u} \\ \sec u &= \frac{1} {\cos u} \\ \cot u &= \frac{1} {\tan u} \end{align}$$

Pythagorean Identities

$$\begin{align} 1 &= \sin^2 u + \cos^2 u \\ \sec^2 u &= 1 + \tan^2 u \\ \csc^2 u &= 1 + \cot^2 u \end{align}$$

Cofunction Identities

$$\begin{align} \sin \left (\frac{\pi} {2} - u \right ) &= \cos u \\ \cos \left (\frac{\pi} {2} - u \right ) &= \sin u \\ \tan \left (\frac{\pi} {2} - u \right ) &= \cot u \\ \csc \left (\frac{\pi} {2} - u \right ) &= \sec u \\ \sec \left (\frac{\pi} {2} - u \right ) &= \csc u \\ \cot \left (\frac{\pi} {2} - u \right ) &= \tan u \end{align}$$

Even-Odd Identities

$$\begin{align} \sin(-u) &= -\sin u \\ \cos(-u) &= \cos u \\ \tan(-u) &= -\tan u \end{align}$$

Sum-Difference Formulas

$$\begin{align} \sin(u \pm v) &= \sin u \cos v \pm \cos u \sin v \\ \cos(u \pm v) &= \cos u \cos v \mp \sin u \sin v \\ \tan(u \pm v) &= \frac{\tan u \pm \tan v} {1 \mp \tan u \tan v} \end{align}$$

Double Angle Formulas

$$\begin{align} \sin(2u) &= 2 \sin u \cos u \\ \cos(2u) &= \cos^2 u - \sin^2 u \\ &= 2 \cos^2 u - 1 \\ &= 1 - 2 \sin^2 u \\ \tan(2u) &= \frac{2 \tan u} {1-\tan^2 u} \end{align}$$

Power-Reducing or Half-Angle Formulas

$$\begin{align} \sin^2 u &= \frac{1-\cos(2u)} {2} \\ \cos^2 u &= \frac{1+\cos(2u)} {2} \\ \tan^2 u &= \frac{1-\cos(2u)} {1+\cos(2u)} \end{align}$$

Sum-to-Product Formulas

$$\begin{align} \label{eq:sum_to_product} \sin u + \sin v &= 2 \sin \frac{u+v} {2} \cos \frac{u - v} {2} \\ \sin u - \sin v &= 2 \cos \frac{u+v} {2} \sin \frac{u - v} {2} \\ \cos u + \cos v &= 2 \cos \frac{u+v} {2} \cos \frac{u-v} {2} \\ \cos u - \cos v &= -2 \sin \frac{u+v} {2} \sin \frac{u-v} {2} \end{align}$$

Product-to-Sum Formulas

$$\begin{align} \label{eq:product_to_sum} \sin u \sin v &= \frac{1} {2} \left [ \cos(u - v) - \cos(u + v) \right ] \\ \cos u \cos v &= \frac{1} {2} \left [ \cos(u - v) + \cos(u + v) \right ] \\ \sin u \cos v &= \frac{1} {2} \left [ \sin(u + v) + \sin(u - v) \right ] \\ \cos u \sin v &= \frac{1} {2} \left [ \sin(u + v) - \sin(u - v) \right ] \end{align}$$

Two-to-One Formulas

$$\begin{align} \label{eq:two_to_one} A \sin u + B \cos u &= C \sin(u + \phi) \\ &= C \cos(u + \psi) \ \text{where} \\ C &= \sqrt{A^2 + B^2} \\ \phi &= \phantom{-}\arctan\frac{B} {A} \\ \psi &= -\arctan\frac{A} {B} \end{align}$$

Online Resources for Section A.2

No online resources.