Type any angle, positive, negative or bigger than a full turn, in degrees or radians (π forms like 7π/4 work). You get the reference angle in both units, the quadrant, the trig values with the right signs, the working step by step and a unit circle sketch.
Last updated:
Degrees or radians: 225, -60, 750, 5π/6, 7pi/4, 2.5
Unit
Enter your values to see the result.
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How to use the Reference Angle Calculator
Type the angle: a number like 225 or -60, or a radian expression such as 5π/6 or 7pi/4.
Choose the unit: degrees or radians. Anything with π is read as radians automatically.
Press Find reference angle; results also update as you type.
Read the reference angle, the quadrant and the coterminal angle between 0° and 360°.
Use the trig table to see how the reference angle gives sin, cos and tan with the correct sign.
Reference angle formulas by quadrant
Start with the angle's coterminal angle θ between 0° and 360° (0 and 2π). The reference angle θ′ is then found from the quadrant that holds the terminal side.
Quadrant
θ in degrees
θ′ (degrees)
θ′ (radians)
Positive functions
I
0° to 90°
θ
θ
all
II
90° to 180°
180° − θ
π − θ
sin, csc
III
180° to 270°
θ − 180°
θ − π
tan, cot
IV
270° to 360°
360° − θ
2π − θ
cos, sec
Worked examples
Angle
Coterminal in [0°, 360°)
Quadrant
Reference angle
135°
135°
II
45° (π/4)
240°
240°
III
60° (π/3)
330°
330°
IV
30° (π/6)
−45°
315°
IV
45° (π/4)
500°
140°
II
40° (2π/9)
7π/6
210°
III
π/6 (30°)
−5π/3
60°
I
π/3 (60°)
3 rad
≈ 171.89°
II
π − 3 ≈ 0.1416 rad (≈ 8.11°)
Tips
Think “distance to the x-axis”. The reference angle is measured to the horizontal axis, never to the y-axis.
Radians without π (like 2.5) are plain numbers: compare them with π ≈ 3.1416, π/2 ≈ 1.5708 and 3π/2 ≈ 4.7124 to find the quadrant.
Sign comes from the quadrant, size from θ′. cos 300° = +cos 60° = 1/2 because cosine is positive in Quadrant IV.
Need more than one coterminal angle? The coterminal angle calculator lists them all.
Frequently asked questions
What is a reference angle?
It is the acute angle between the terminal side of an angle and the x-axis. It is always between 0° and 90° (0 and π/2) and is never negative. For example, the reference angle of 150° is 30°, because the terminal side is 30° away from the negative x-axis.
How do you find a reference angle?
First bring the angle into the range 0° to 360° by adding or subtracting 360° as needed. Then use the quadrant: in Quadrant I it is the angle itself, in Quadrant II it is 180° − θ, in Quadrant III it is θ − 180°, and in Quadrant IV it is 360° − θ.
How do you find a reference angle in radians?
Use the same rules with π instead of 180°: Quadrant II gives π − θ, Quadrant III gives θ − π and Quadrant IV gives 2π − θ. For 5π/6 (Quadrant II) the reference angle is π − 5π/6 = π/6.
What is the reference angle of a negative angle?
Add 360° (or 2π) until the angle is positive, then apply the quadrant rule. −60° is coterminal with 300°, which is in Quadrant IV, so its reference angle is 360° − 300° = 60°.
What is the reference angle of 90°, 180° or 270°?
These are quadrantal angles: their terminal side lies on an axis, so they are not in any quadrant. Measured to the nearest part of the x-axis, 180° and 360° have a reference angle of 0°, while 90° and 270° have a reference angle of 90°.
Why are reference angles useful?
Trig functions of any angle equal the same function of its reference angle, up to a sign set by the quadrant. So sin 210° = −sin 30° = −1/2. This lets you evaluate any angle using only the values from 0° to 90°.
What does ASTC (All Students Take Calculus) mean?
It is a memory aid for which trig functions are positive in each quadrant: All in Quadrant I, Sine in Quadrant II, Tangent in Quadrant III and Cosine in Quadrant IV. Their reciprocals (csc, cot, sec) share the same signs.
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