Trigonometry

Definition of trigonometric functions

Trigonometry studies relationships between angles and side lengths, especially in right triangles and circles. It is essential in Geometry, Vector Calculus index, waves, rotations, and projections.

For a right triangle with angle $\theta$:

$$\begin{array}{rl}\sin \theta & =\frac{opp}{hyp}\\ \csc \theta & =\frac{hyp}{opp}=\frac{1}{\sin \theta }\\ \cos \theta & =\frac{adj}{hyp}\\ \sec \theta & =\frac{hyp}{adj}=\frac{1}{\cos \theta }\\ \tan \theta & =\frac{opp}{adj}=\frac{\sin \theta }{\cos \theta }\\ \cot \theta & =\frac{adj}{opp}=\frac{\cos \theta }{\sin \theta }=\frac{1}{\tan \theta }\end{array}$$

On the unit Circle, $\cos \theta$ is the $x$-coordinate and $\sin \theta$ is the $y$-coordinate of the point reached by rotating angle $\theta$ from the positive $x$-axis. This makes trig functions periodic and links them directly to rotation.

Identities and Laws

$$\begin{array}{rl}{\sin }^2\theta +{\cos }^2\theta & =1\\ {\sec }^2\theta & =1+{\tan }^2\theta \\ {\csc }^2\theta & =1+{\cot }^2\theta \end{array}$$

Law of sines:

$$\frac{\sin \alpha }{a}=\frac{\sin \beta }{b}=\frac{\sin \gamma }{c}$$

Law of cosines:

$$\begin{array}{rl}{a}^2& =b^2+c^2-2bc\cos \alpha \\ b^2& =a^2+c^2-2ac\cos \beta \\ c^2& =a^2+b^2-2ab\cos \gamma \end{array}$$

The dot product formula $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$ shows why trigonometry is the language of angles between vectors in Euclidean spaces.