# Trigonometry

## Angles

### Naming angles

#### Building an angle

Each side of an **angle** is bounded by a **ray** (a directed line segment). When we put two rays together with a common endpoint, the endpoint is called the **vertex** of the angle, and the two rays are the sides of the angle.

There are different systems to measure angles. Typically, **degrees** and **radians** are used. $360$ degrees is equivalent to $2\pi$ radians.

Greek letters are typically used as variables for the measure of an angle. Theta $\theta$ is highly common.

#### Types of angles

Angle in degrees |
Angle in radians |
Angle name |

$\theta = 0\degree$ |
$\theta = 0$ |
Zero angle |

$0\degree < \theta < 90\degree$ |
$0 < \theta < \frac{\pi}{2}$ |
Acute angle |

$\theta = 90\degree$ |
$\theta = \frac{\pi}{2}$ |
Right angle |

$90\degree < \theta < 180\degree$ |
$\frac{\pi}{2} < \theta < \pi$ |
Obtuse angle |

$\theta = 180\degree$ |
$\theta = \pi$ |
Straight angle |

$180\degree < \theta < 360\degree$ |
$\pi < \theta < 2\pi$ |
Reflex angle |

$\theta = 360\degree$ |
$\theta = 2\pi$ |
Complete angle |

### Complimentary and supplementary angles

**Complimentary** angles are two angles that sum to $90\degree$ or $\frac{\pi}{2}$.

**Supplementary** angles are two angles that sum to $180\degree$ or $\pi$.

## Trigonometric identities

### Basic identities

- $\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} = \ln(\cos{\theta})$
- $\cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} = \ln(\sin{\theta})$

### Pythagorean identities

The fundamental Pythagorean Trigonometric identity is:

$\sin x + \cos x = 1$

From this formula we can derive the formulas for other functions:

Sine and cosine:

$\sin^2 x + \cos^2 x = 1$

- $\cos^2 x = 1 - \sin^2 x$
- $\sin^2 x = 1 - \cos^2 x$

Tangent and secant:

$1 + \tan^2 x = \sec^2 x$

- $\tan^2 x = \sec^2 x - 1$
- $1 = \sec^2 x - \tan^2 x$

Cotangent and cosecant:

$1 + \cot^2 x = \csc^2 x$

- $\cot^2 x = \csc^2 x - 1$
- $1 = \csc^2 x - \cot^2 x$