# Sin, Cos, and Tan

### Sin, Cos, and Tan

#### Introduction

• Given a right angled triangle:
• We have labelled:
• The angle as θ.
• The hypotenuse as h, which is the longest side, and is opposite the right angle.
• The adjacent side as a, which is next to the angle.
• The opposite side as o, which is opposite the angle.
• sin, cos, and tan are functions which convert the angle to a ratio of two of the sides:

#### Example

• Given the triangle:
• We can calculate:

#### Code (Python)

import math

# python works in radians, not degrees

print(math.sin(theta)) # prints 0.5
print(math.cos(theta)) # prints 0.866...
print(math.tan((theta))) # prints 0.577...

#### Notes

• sin, cos, and tan are sometimes called sine, cosine, and tangent.
• A simple way to remember the equations for sin, cos, and tan is sohcahtoa:
• soh: sin opposite hypotenuse

#### tan as a function of sin and cos

• If we re-arrange the equations for the opposite and adjacent:
• Then we can rewrite tan in terms of sin and cos:
• Which simplifies to:

#### Graphs of sin, cos, and tan

• We can plot sin on a graph, where:
• x is between 0 and 360
• y = sin(x)
• We see that sin(x) is always between -1 and 1
• And for cos:
• We see that cos(x) is just sin(x) shifted left by 90°
• And finally tan:
• We see that tan(90) and tan(270) aren't defined, as cos(90) and cos(270) are 0, which causes a divide by 0 error.