Math · Trigonometry
Unit circle calculator.
Enter any angle in degrees or radians. In radians you can type pi/6, 5pi/4 or 2pi/3. The point on the circle, all six trig values and the exact surd forms at special angles are shown.
At 150° (5π/6) the point on the unit circle is (cos, sin) = (−√3/2, 1/2) ≈ (−0.866025, 0.5). So sin 150° = 1/2, cos 150° = −√3/2 and tan 150° = −√3/3. The angle is in Quadrant II with a reference angle of 30°.
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How the unit circle works
The unit circle has radius 1 and its centre at the origin. An angle θ measured from the positive x-axis meets the circle at the point (cos θ, sin θ), so cosine is the x-coordinate and sine is the y-coordinate. Tangent is sin θ / cos θ, and cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. Where the reciprocal's denominator is 0 the value is undefined.
Exact values come from the special angles 30° (π/6), 45° (π/4) and 60° (π/3): sin is 1/2, √2/2 and √3/2 and cos is √3/2, √2/2 and 1/2. Every other angle that is a multiple of 30° or 45° uses the same numbers, with signs set by the quadrant and the reference angle (the acute angle to the x-axis).
Worked example
150° is in Quadrant II with reference angle 180° − 150° = 30°. Sine is positive there and cosine negative, so sin 150° = 1/2 and cos 150° = −√3/2, giving tan 150° = −√3/3 and a point of about (−0.866025, 0.5). For 7π/6 (210°, Quadrant III, reference angle 30°) both are negative: sin = −1/2, cos = −√3/2 and tan = √3/3.
Limits
Angles of any size are reduced to the coterminal angle between 0° and 360°. Decimal answers are rounded to six places. Exact surd forms are shown only at multiples of 30° and 45°.
Sources, opened 15 October 2026: OpenStax, Precalculus 2e, 5.2 Unit Circle: Sine and Cosine Functions (sine and cosine of the special angles 30°, 45° and 60°).
Frequently asked questions
What is the unit circle?
A circle of radius 1 centred at the origin. The point where an angle θ meets it is (cos θ, sin θ), which is why the unit circle is used to define sine and cosine for all angles.
How do I find sin and cos of 150 degrees?
150° has a reference angle of 30° in Quadrant II, where sine is positive and cosine is negative. So sin 150° = 1/2 and cos 150° = −√3/2.
Can I enter radians with pi?
Yes. Choose Radians and type values such as pi/6, 5pi/4, 2pi/3 or a decimal like 0.5236. The calculator converts to degrees first.
What is a reference angle?
The acute angle between the terminal side of the angle and the x-axis. It lets you reuse the values from 0° to 90° in every quadrant, with the sign set by the quadrant.
Why is tan 90 degrees undefined?
Because cos 90° = 0 and tangent is sin / cos, which would divide by zero. The same is true at 270°.
Learning aid
Educational calculator with worked steps. Check answers and the required method against your own course or textbook. Nothing you type leaves your browser.