Section 1.4 Reference Angles

Section Objectives

  1. Find reference angles.
  2. Use the reference angle to evaluate the trig functions at any angle.


Reference Angles

Even though the right triangle definitions of the trig functions require that is an acute angle, that requirement can easily be eliminated by using reference angles.



Let be an angle in standard position. Its reference angle is the acute angle formed by the terminal side of and the horizontal axis.

 

 



Examples

Find the reference angle for .




Find the reference angle for .




Find the reference angle for .




Signs of the Trig Functions in the Quadrants

Based on our unit circle definitions of the trig functions, the following chart should be pretty clear. And these ideas are worth getting used to!




Evaluating the Trig Functions at any Angle

To evaluate a trigonometric function at any angle :

  1. Determine the reference angle for . Let's call it .
  2. Evaluate the trig function at .
  3. Depending on the quadrant in which lies, give the appropriate sign to the value you found in step 2.



Examples

Evaluate .




Evaluate .