# 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:

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