We want to consider how to evaluate the trigonometric ratios of angles in the four quadrants. When evaluating the trigonometric ratios of nonacute angles, we need to consider the concept of reference angles.
The table below shows the reference angle, α , in quadrant I which corresponds to the angle, θ , in quadrants II, III, and IV.
Quadrant 
Angle θ 
Reference angle α 
Diagram 
II 
90˚ < θ < 180 ˚ 
α = 180 ˚ – θ 

III 
180˚ < θ < 270 ˚ 
α = θ – 180˚ 

IV 
270˚ < θ < 360 ˚ 
α = 360 ˚ – θ 
Determine the reference angle that corresponds to each of the following angle.
a) 165˚
b) 249˚
c) 328˚
a) 165˚ is in quadrant II (90˚ < 165˚ < 180˚ )
The reference angle is 180˚ – 165˚ = 15˚
b) 249˚ is in quadrant III (180˚ < 249˚ < 270˚ )
The reference angle is 249˚ – 180˚ = 69˚
c) 328˚ is in quadrant III (270˚ < 328˚ < 360˚ )
The reference angle is 360˚ – 328˚ = 32˚
A discussion of what reference angles are and how to find them, and then how to use them to determine the sine and cosine values of angles greater than ninety degrees.
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with stepbystep explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.