What is a reference angle?

A reference angle is the acute angle between 0 and 90 degrees formed between the terminal side of an angle and the x-axis. It lets you work out the trig functions of any angle from the values of a familiar acute angle, adjusting the sign for the quadrant.

How do you find the reference angle for 150 degrees?

150 degrees is in the second quadrant, between 90 and 180 degrees. The reference angle is 180 minus 150 = 30 degrees. Sine, cosine and tangent of 150 degrees have the same size as those of 30 degrees, with signs set by the second quadrant.

How do you find reference angles in general?

First reduce the angle to between 0 and 360 degrees. In quadrant 1 the reference angle is the angle itself. In quadrant 2 it is 180 minus the angle. In quadrant 3 it is the angle minus 180. In quadrant 4 it is 360 minus the angle.

Reference Angle Calculator

The reference angle is one of the most useful ideas in trigonometry. It is the acute angle, always between 0 and 90 degrees, formed between the terminal side of any angle and the nearest part of the x-axis. Its power comes from the symmetry of the unit circle: the size of the sine, cosine, and tangent of any angle is exactly the same as for its reference angle. Only the sign changes depending on the quadrant the original angle sits in. So once you know that 150 degrees has a reference angle of 30 degrees, you immediately know that the sine and cosine of 150 degrees have the same size as those of 30 degrees, and you only need to check the sign rules for the second quadrant. This calculator takes any angle, positive or negative, in degrees or radians, and returns the reference angle and the quadrant. It first reduces your angle to the standard 0 to 360 degree range by adding or subtracting full turns as needed, which handles very large angles and negative angles without any extra steps from you. Then it applies the right rule for the quadrant. The coterminal angle in the 0 to 360 range is shown too, so you can see exactly where the original angle lands before the reference angle is read off. Use this tool for senior school and university trigonometry, evaluating trig functions of awkward angles, simplifying expressions, and preparing for tests where the unit circle matters. It also helps you develop a feel for how angles relate across all four quadrants.

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Angle

Unit

30°

reference angle

QuadrantQ2

Coterminal (0 to 360)150°

How it works

The angle is first reduced to between 0 and 360 degrees by adding or subtracting full turns. Then: in quadrant 1 (0 to 90) the reference angle is the angle itself; in quadrant 2 (90 to 180) it is 180 minus the angle; in quadrant 3 (180 to 270) it is the angle minus 180; in quadrant 4 (270 to 360) it is 360 minus the angle. The result is always between 0 and 90 degrees.

Worked example

For 150 degrees: the angle is already in the 0 to 360 range and sits in the second quadrant, between 90 and 180. The reference angle is 180 minus 150 = 30 degrees in quadrant 2. So sin(150) = sin(30) = 0.5 (positive in Q2), cos(150) = -cos(30) = -0.866 (negative in Q2), and tan(150) = -tan(30) (negative in Q2). These match the pre-filled defaults above.

Related calculators

Unit Circle Calculator: coordinates, sine, cosine and tangent at any angle.
Inverse Trig Calculator: find arcsin, arccos and arctan from a ratio.
SOHCAHTOA Calculator: solve right triangles using trig ratios.
Law of Sines and Cosines: solve non-right triangles.

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