Coterminal & Reference Angles

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In mathematics, coterminal and reference angles play crucial roles in trigonometry. Coterminal angles have the same terminal side, while reference angles help simplify calculations. Discover formulas, examples, and practical applications in this comprehensive guide.

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Uploaded on Apr 16, 2025 | 1 Views

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Reference and Coterminal Angles

What are coterminal angles? These are two or more angles in standard position with the same terminal side. Standard position is where the vertex is at the origin and the initial side along the positive x-axis. These angles are found by taking the measure of the angle, and adding multiples of 360 degrees or 2 . Every angle has infinite coterminal angles.

Coterminal angles example Image source: Coterminal Angles

What is a reference angle? This is the acute angle formed by the terminal side of the given angle and the X-axis. This is only if the angle is not one of the four quadrantal angles (90 , 180 , 270 , 360 ). If the angle is greater than 360 , or less than 0 , the reference angle can be found by using a coterminal angle.

Reference angle example Image source: Reference Angles

Reference Angle Formulas Quadrant 1 = Actual Quadrant 2 = 180 Angle in radians = Angle Quadrant 3 = Angle 180 in radians = Angle Quadrant 4 = 360 angle in radians = 2 angle

Example 1 Q: If the angle measure is 50 in standard position, find one coterminal angle. A: 50 + 360 = 410

Example 2 Q: If the angle measure is 110 in standard position, find the reference angle. A: 180 110 = 70

Example 3 Q: If the angle measure is 120 , find the one coterminal angle and reference. A: 120 + 360 = 240 (coterminal) 240 180 = 60 (reference)

Example 4 Q: If the angle measure is 279 , find two coterminal angles and the reference angle. A: 279 + 360 = 639 (coterminal) 279 + (360(2)) = 999 (coterminal) 360 279 = 81 (reference)