College Algebra & Trigonometry, 2018a

9.1. TRIGONOMETRIC FUNCTIONS OF NON-ACUTE ANGLES 391
Reference Angles for Negative Angles
Negatively measured angles rotate in a clockwise direction.
θ + 180 ◦
θ
1
(cos θ, sin θ)
There are a variety of methods for finding the reference angle for a negatively
valued angle. You can find a positive angle that is co-terminal with the negative
angle and then find the reference angle for the positive angle. You can also drop
a perpendicular to the x-axis to find the reference angle for the negative angle
directly.
For example, the angle −120 ◦ terminates in Quadrant III and is co-terminal with
the positive angle 240 ◦ . Either way, when you drop a perpendicular to the x-axis,
you find that the reference angle is 60 ◦ .
If you are given the value of one of the trigonometric functions of an angle θ,
and know which quadrant θ is located in, you can find the other trigonometric
functions for that angle.
Example
Given θ in Quadrant IV with cos θ = 1 5
, find sin θ and tan θ.
If cos θ = 1 5
, then the adjacent side and the hypotenuse must be in a ratio of 1:5.