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
theta = math.radians(30)

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
  • cah: cos adjacent hypotenuse
  • toa: tan opposite adjacent

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.