We already know how to find the coterminal angles of a given angle. The reference angle of any angle always lies between 0° and 90°, It is the angle between the terminal side of the angle and the x-axis. The reference angle depends on the quadrant’s terminal side.
The steps to find the reference angle of an angle depends on the quadrant of the terminal side:
- We first determine its coterminal angle which lies between 0° and 360°.
- We then see the quadrant of the coterminal angle.
- If the terminal side is in the first quadrant ( 0° to 90°), then the reference angle is the same as our given angle. For example, if the given angle is 25°, then its reference angle is also 25°.
- If the terminal side is in the second quadrant ( 90° to 180°), then the reference angle is (180° – given angle). For example, if the given angle is 100°, then its reference angle is 180° – 100° = 80°.
- If the terminal side is in the third quadrant (180° to 270°), then the reference angle is (given angle – 180°). For example, if the given angle is 215°, then its reference angle is 215° – 180° = 35°.
- If the terminal side is in the fourth quadrant (270° to 360°), then the reference angle is (360° – given angle). For example, if the given angle is 330°, then its reference angle is 360° – 330° = 30°.
Example: Find the reference angle of 495°.
Solution:
Let us find the coterminal angle of 495°. The coterminal angle is 495° − 360° = 135°.
The terminal side lies in the second quadrant. Thus the reference angle is 180° -135° = 45°
Therefore, the reference angle of 495° is 45°.
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