Reference Angle Calculator
This reference angle calculator finds the acute angle that the terminal side of an angle makes with the x-axis. Example: the reference angle of 210° is 30°, because 210° − 180° = 30°.
Example: Unit: Degrees (°) · Angle 210 → Result: Reference angle = 30° (angle 210°, Quadrant III). Source: Wikipedia – Reference angle. Updated: .
How it is calculated
The reference angle is the acute angle between the terminal side of an angle and the x-axis. It lets you reduce sine, cosine and tangent values to familiar angles between 0° and 90°.
Formula and steps
First bring the angle θ into the range 0° to 360°. Then the formula depends on the quadrant: in the first it is θ itself, in the second 180° − θ, in the third θ − 180° and in the fourth 360° − θ. If the angle lies exactly on an axis, the reference angle is 0° (at 0° and 180°) or 90° (at 90° and 270°). Convert radians to degrees first with 180° / π.
Worked examples
210° lies in the third quadrant: 210° − 180° = 30°. For 135° in the second quadrant, 180° − 135° = 45°. The angle −45° equals 315° and gives 360° − 315° = 45° in the fourth quadrant. For 400° you first subtract 360°, leaving 40°. In radians, 5.235988 rad equals 300°, so the reference angle is 60°.
- With the reference angle 30°, sin 210° = −sin 30° = −0.5; the minus sign comes from the third quadrant.
- The reference angle is not the same as a coterminal angle, which is only shifted by full turns.
Frequently asked questions
What is a reference angle?
The acute angle between the terminal side and the x-axis. It always lies between 0° and 90° and gives the absolute value of the trigonometric functions.
How do I find the reference angle of 210°?
210° is in the third quadrant, so 210° − 180° = 30°.
What about negative angles or angles above 360°?
Add or subtract full turns of 360° until the angle lies between 0° and 360°. −45° becomes 315° and has the reference angle 45°.
Sources and legal basis
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