Related Angles and the Quadrants

This page will explain what related angles are and how they are defined in the four parts of the coordinate axes (quadrants).

A Related Angle, also called a Reference Angle, of a given angle is the positive Acute Angle (an angle less than 90°) between the x-axis and the terminal side of the given angle.

Any trigonometric function of an angle is numerically equal to the same function of its Related Angle.

The sign of the trigonometric functions of the Related Angle depends on which Quadrant the original angle is located.

To Find the Related (Acute) Angle of θ: Quadrant Condition Formula 1 0° <= θ <= 90° Related angle = θ 2 90° < θ <= 180° Related angle = 180° - θ 3 180° < θ <= 270° Related angle = θ - 180° 4 270° < θ <= 360° Related angle = 360° - θ Radian - Degree Degree Radian (fraction) Radian (decimal) 0° 0 0 90° π/2 ≈ 1.570796327 180° π ≈ 3.141592654 270° 3π/2 ≈ 4.71238898 360° 2π ≈ 6.283185307 Sign of the Cosine and Sine Functions: Quadrant Cosine Function Sine Function 1 Positive value Positive value 2 Negative value Positive value 3 Negative value Negative value 4 Positive value Negative value