Math
Reference Angle Calculator
A reference angle is the acute angle (0° to 90°) between an angle's terminal side and the x-axis.
It lets you evaluate any trig function from the first-quadrant values you already know. Enter any angle, in degrees or radians, negative or larger than a full turn, and get the reference angle, quadrant, coterminal angles, and the trig values that follow.
Type a number, a fraction like 180/7, a π expression like 5π/6, or degrees-minutes-seconds like 30° 15′ 30″. Negative and large angles are fine.
Try an example
Result
Reference angle
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- Quadrant
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- Coterminal in [0°, 360°)
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- sin θ
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- cos θ
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Reference angle rules by quadrant
| Quadrant | θ range | Reference angle (degrees) | Reference angle (radians) |
|---|---|---|---|
| I | 0° to 90° | θ | θ |
| II | 90° to 180° | 180° − θ | π − θ |
| III | 180° to 270° | θ − 180° | θ − π |
| IV | 270° to 360° | 360° − θ | 2π − θ |
Why reference angles matter
Sine, cosine, and tangent of an angle equal the same function of its The small positive angle between a given angle's arm and the x-axis., except for a sign. So sin 225° = -sin 45° = -√2/2 because 225° is in One of the four regions the x- and y-axes cut the coordinate plane into. III, where sine is negative. Memorize the first-quadrant values and reference angles do the rest.
How to use it
- Enter the angle in degrees or An angle unit where a full turn is 2π instead of 360 degrees. (π fractions such as 7π/6 are fine).
- Read the reference angle in the header and the quadrant rule that produced it.
- Use the sin, cos, and tan values to check your own work.
How to read the answer
- The reference angle is always between 0° and 90°, inclusive.
- Quadrantal angles (0°, 90°, 180°, 270°) sit on an axis; their reference angle is 0° or 90°.
- Negative angles are first converted to their positive coterminal angle: -30° becomes 330°, whose reference angle is 30°.
Common mistakes and edge cases
- Subtracting from 360° in Quadrant III: the Quadrant III rule is θ − 180°, not 360° − θ.
- Forgetting to reduce first: 1000° is not in any quadrant until you subtract 720° to get 280°.
- Dropping the quadrant sign afterwards: the reference angle gives the size of the trig value, the quadrant gives the sign.
- Reading radians as degrees: 7π/6 is 210°, in Quadrant III, with reference angle π/6.
Worked examples
Quadrant III angle
225°
Reference angle
45° = π/4 (0.785398)
Negative angle
-30° is coterminal with 330°
Reference angle
30° = π/6 (0.523599)
Radians
7π/6
Reference angle
30° = π/6 (0.523599)
More than one turn
1000° reduces to 280°
Reference angle
80° = 4π/9 (1.396263)
Quadrantal angle
180° lies on the x-axis
Reference angle
0° = 0
Decimal angle
123.4°
Reference angle
56.6° = 283π/900 (0.987856)
Unreadable input
Reference angle
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Frequently asked questions
What is the reference angle of 225°?+
45°. 225° is in Quadrant III, so subtract 180°: 225° − 180° = 45°.
What is the reference angle of a negative angle?+
Add 360° until the angle is positive, then apply the quadrant rule. -30° + 360° = 330°, which is in Quadrant IV, so the reference angle is 360° − 330° = 30°.
Can a reference angle be 0° or 90°?+
Yes, for angles on the axes. 0°, 180°, and 360° have reference angle 0°; 90° and 270° have reference angle 90°.
What is the reference angle in radians?+
The same rules with π in place of 180°: Quadrant II uses π − θ, Quadrant III uses θ − π, Quadrant IV uses 2π − θ. For 7π/6 the reference angle is 7π/6 − π = π/6.
Is the reference angle the same as the coterminal angle?+
No. A coterminal angle differs from θ by a whole number of turns and can be any size. The reference angle is always acute and is measured to the x-axis.
How do I use the reference angle to find sin 300°?+
300° is in Quadrant IV, reference angle 60°. sin 60° = √3/2 and sine is negative in Quadrant IV, so sin 300° = -√3/2.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026