Reference Angle

 In this lesson, we will look into how to find the reference angle.



The Cartesian plane is divided into 4 quadrants by the two coordinate axes. These 4 quadrants are labelled I, II, III and IV respectively.




We want to consider how to evaluate the trigonometric ratios of angles in the four quadrants. When evaluating the trigonometric ratios of non-acute 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.



Angle θ

Referemce angle α



90˚ < θ < 180 ˚

α = 180 ˚ – θ


180˚ < θ < 270 ˚

α = θ – 180˚


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˚




Sine and cosine at non-acute angles
To find the value of sine and cosine at non-acute angles (from 90 to 360), first draw the angle on the unit circle and find the reference angle. A reference angle is formed by the terminal side and the x-axis and will therefore always be acute. When evaluating cosine and sine for the reference angle, determine if each value is positive or negative by identifying the quadrant the terminal side is in.

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.




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