Reference Angle
How to find the reference angle of the given angle: formula, 3 examples, and their solutions.
Formula
The reference angle is the smaller angle
formed by the x-axis and the terminal side.
If θ is in quadrant I,
then the reference angle is
θ.
If θ is in quadrant II,
then the reference angle is
π - θ.
If θ is in quadrant III,
then the reference angle is
θ - π.
If θ is in quadrant IV,
then the reference angle is
2π - θ.
Example2π/3
2π/3 is between π/2 and π.
So draw the terminal side on quadrant II.
Draw the reference angle.
It's the smaller angle
formed by the x-axis and the terminal side.
Then the reference angle is
π - 2π/3.
π = 3π/3
3π/3 - 2π/3 = π/3
So π/3 is the answer.
Example5π/4
5π/4 is between π and 3π/2.
So draw the terminal side on quadrant III.
Draw the reference angle.
It's the smaller angle
formed by the x-axis and the terminal side.
Then the reference angle is
5π/4 - π.
-π = -4π/4
5π/4 - 4π/4 = π/4
So π/4 is the answer.
Example11π/6
11π/6 is between 3π/2 and 2π.
So draw the terminal side on quadrant IV.
Draw the reference angle.
It's the smaller angle
formed by the x-axis and the terminal side.
Then the reference angle is
2π - 11π/6.
2π = 12π/6
12π/6 - 11π/6 = π/6
So π/6 is the answer.