REFERENCE ANGLES

Topics in this lesson:
1. THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES
2. THE REFERENCE ANGLE OF THE SPECIAL ANGLES
3. USING A REFERENCE ANGLE TO FIND THE VALUE OF THE SIX
TRIGONOMETRIC FUNCTIONS
1. THE DEFINITION AND EXAMPLES OF REFERENCE ANGLES
Definition The reference angle of the angle ? , denoted by ? , is the acute angle
determined by the terminal side of ? and either the positive or negative x-axis.
Recall, an acute angle is an angle whose measurement is greater than 0? and less
than 90? .
NOTE: By definition, the reference angle is an acute angle. Thus, any angle
whose terminal side lies on either the x-axis or the y-axis does not have a reference
angle.
Examples Find the reference angle ? for the following angles ? made by rotating
counterclockwise.
1. ? is in the I quadrant.
Animation of the reference angle.
?
?
Relationship between ? and ? : ? ? ?
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
2. ? is in the II quadrant.
Animation of the reference angle.
? ?
Relationship between ? and ? :
? ? ? ? ? OR ? ? 180? ? ?
OR
? ? ? ? ? OR ? ? ? ? 180?
3. ? is in the III quadrant.
Animation of the reference angle.
?
?
Relationship between ? and ? :
? ? ? ? ? OR ? ? ? ? 180?
OR
? ? ? ? ? OR 180? ? ? ? ?
Copyrighted by James D. Anderson, The University of Toledo
www.math.utoledo.edu/~janders/1330
4. ? is in the IV quadrant.
Animation of the reference angle.
?
?
Relationship between ? and ? :
? ? 2? ? ? OR ? ? 360? ? ?
OR
? ? ? ? 2? OR ? ? ? ? 360?
Examples Find the reference angle ? for the following angles ? made by rotating
clockwise.
1. ? is in the IV quadrant.
Animation