# 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.

## Basic definitions

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

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

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

### Sine

The sine of an angle $\theta$, abbreviated $\sin \theta$, is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\sin 30^{\circ}=\frac 12$.

### Cosine

The cosine of an angle $\theta$, abbreviated $\cos \theta$, is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\cos 30^{\circ} =\frac{\sqrt{3}}{2}$.

### Tangent

The tangent of an angle $\theta$, abbreviated $\tan \theta$, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\tan 30^{\circ}=\frac{\sqrt{3}}{3}$. (Note that $\tan \theta=\frac{\sin\theta}{\cos\theta}$.)

### Cosecant

The cosecant of an angle $\theta$, abbreviated $\csc \theta$, is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\csc 30^{\circ}=2$. (Note that $\csc \theta=\frac{1}{\sin \theta}$.)

### Secant

The secant of an angle $\theta$, abbreviated $\sec \theta$, is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\sec 30^{\circ}=\frac{2\sqrt{3}}{3}$. (Note that $\sec \theta=\frac{1}{\cos \theta}$.)

### Cotangent

The cotangent of an angle $\theta$, abbreviated $\cot \theta$, is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\cot 30^{\circ}=\sqrt{3}$. (Note that $\cot \theta=\frac{\cos\theta}{\sin\theta}$ or $\cot \theta = \frac{1}{\tan \theta}$.)

## Trigonometry 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 $\theta$ and the coordinates of the point we picked to be $(x,y)$, 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*}

Note that $(x,y)$ is the rectangular coordinates for the point $(1,\theta)$.

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.

### Pythagorean Identities

• $\sin^2\theta+\cos^2\theta=1$
• $1+\tan^2\theta=\sec^2\theta$
• $1+\cot^2\theta=\csc^2\theta$

### Double-Angle Identities

• $\sin 2\theta=2\sin\theta\cos\theta$
• $\cos 2\theta=\cos^2\theta-\sin^2\theta$
• $\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$

### Half-Angle Identities

• $\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$
• $\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}$
• $\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$