# Trigonometry

Trigonometry seeks to find the lengths of a triangle's sides, given 2 angles and a side. Trigonometry is closely related to analytic geometry.

## Contents

## Basic definitions

Usually we call an angle , read "theta", but is just a variable. We could just as well call it .

For the following definitions, the "opposite side" is the side opposite of angle , and the "adjacent side" is the side that is part of angle , but is not the hypotenuse.

i.e. If ABC is a right triangle with right angle C, and angle A = , then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.

### Sine

The sine of an angle , abbreviated , is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .

### Cosine

The cosine of an angle , abbreviated , is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, .

### Tangent

The tangent of an angle , abbreviated , is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)

### Cosecant

The cosecant of an angle , abbreviated , is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)

### Secant

The secant of an angle , abbreviated , is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that .)

### Cotangent

The cotangent of an angle , abbreviated , is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, . (Note that or .)

## Trigonometery Definitions for non-acute angles

Consider a unit circle that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a hypotenuse 1 unit long. Letting the angle at the origin be and the coordinates of the point we picked to be , we have:

\begin{align*} \sin \theta &= y \\ \cos \theta &= x \\ \tan \theta &= \frac{y}{x} \\ \csc \theta &= \frac{1}{y} \\ \sec \theta &= \frac{1}{x} \\ \cot \theta &= \frac{x}{y} \end{align*} (Error compiling LaTeX. ! Paragraph ended before \align* was complete.)

Note that is the rectangular coordinates for the point .

This is true for all angles, even negative angles and angles greater than 360 degrees. Due to the way trig ratios are defined for non-acute angles, the value of a trig ratio could be positive or negative, or even 0.

## Trigonometric Identities

There are many identities that are based on trigonometric functions.