# Reference Angles

Reference angles are used to make the calculation of the trigonometric ratios easier. The values of the trigonometric functions for angle θ will be the same numerical value as the trigonometric values for the reference angle.

A reference angle is the smallest angle that is formed by the x- axis and the terminal side of the angle θ. Therefore, the reference angle is always coterminal with the original angle θ. Reference angles will always have a value between 0° and 90°. In the diagram below θ represents the original angle and R represents the Reference Angle

Same as θ 180 - θ

θ = 45° and R = 45°θ = 135° and R = 180° - 135° = 45°

Quad III Reference AngleQuad IV Reference Angle
θ - 180360 - θ

θ = 225° and θ = 315° and
R = 225 - 180 = 45               R = 360 - 315 = 45

The reference angle for any angle θ in standard position is the positive acute angle between the terminal side of θ and the x-axis. The reference angle is always positive and always between 0° and 90°

 Related Links: Math Trigonometry Coterminal Angles Functions of Large and Negative Angles

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