About the Reference Angle Calculator
The reference angle is the acute angle between an angle’s terminal side and the x-axis. It is always between 0° and 90° (0 and π/2 radians), and it is the key to evaluating trigonometric functions of any angle: sin 150° has the same size as sin 30°, and only the sign depends on the quadrant.
This reference angle calculator accepts any angle — negative, larger than 360°, or in radians — reduces it to its principal angle between 0° and 360°, identifies the quadrant (or the axis it lies on), and returns the reference angle along with the signs of sine, cosine and tangent in that quadrant (the “All Students Take Calculus” rule). It also shows the actual sin, cos and tan values so you can check your work.
Use it for trigonometry and precalculus homework, unit-circle practice, or whenever you need to reduce an angle before looking up an exact value.
With the default inputs, the reference angle is 30. Change any value above to recalculate instantly.
How to use the reference angle calculator
- 1Enter the angle, including a minus sign if it is negative.
- 2Choose degrees or radians.
- 3Read the reference angle and the quadrant.
- 4Use the sign rule to attach + or − to sin, cos and tan of the reference angle.
Formula and method
First the angle is reduced to its principal angle θ between 0° and 360° by adding or subtracting full turns (radians are converted to degrees for this step). The quadrant follows from where θ falls: 0–90° is Quadrant I, 90–180° Quadrant II, 180–270° Quadrant III and 270–360° Quadrant IV.
The reference angle θ′ is the distance from the terminal side to the nearest part of the x-axis, given by the formulas above (in radians use π and 2π instead of 180° and 360°). Angles that land exactly on an axis are quadrantal: their reference angle is 0° on the x-axis and 90° on the y-axis.
- θ
- Principal angle, 0° ≤ θ < 360°
- θ′
- Reference angle, 0° ≤ θ′ ≤ 90°
Worked examples
Reference angle of 150°
150° lies in Quadrant II, so the reference angle is 180° − 150° = 30°. Sine is positive and cosine negative there, so sin 150° = 0.5 and cos 150° ≈ −0.866.
A negative angle: −225°
Adding 360° gives the principal angle 135°, which is in Quadrant II. Its reference angle is 180° − 135° = 45°.
4 radians
4 rad is between π ≈ 3.1416 and 3π/2 ≈ 4.7124, so it is in Quadrant III. The reference angle is 4 − π ≈ 0.8584 rad, or about 49.18°.
300°
300° is in Quadrant IV, so the reference angle is 360° − 300° = 60°. Tangent is negative there: tan 300° = −tan 60° ≈ −1.7321.
Frequently asked questions
What is a reference angle?+
It is the acute angle (0° to 90°) formed between the terminal side of an angle in standard position and the x-axis. It is always measured to the x-axis, never the y-axis.
How do you find the reference angle of a negative angle?+
Add 360° (or 2π) until the angle is between 0° and 360°, then apply the quadrant rule. For −30°, the principal angle is 330°, so the reference angle is 30°.
Can a reference angle be negative or larger than 90°?+
No. A reference angle is always between 0° and 90° inclusive (0 to π/2 radians), because it measures the shortest distance to the x-axis.
How do I find the reference angle in radians?+
Use the same rules with π: Quadrant II θ′ = π − θ, Quadrant III θ′ = θ − π, Quadrant IV θ′ = 2π − θ. For 5π/4 the reference angle is π/4.
What does “All Students Take Calculus” mean?+
It is a memory aid for which functions are positive: All in Quadrant I, Sine in II, Tangent in III and Cosine in IV. The reference angle gives the size; the quadrant gives the sign.