Unit Circle Calculator

The unit circle is the circle of radius 1 centered at the origin. Draw a ray from the origin at angle θ, measured counterclockwise from the positive x-axis, and it meets the circle at one point. That point is (cos θ, sin θ). Every other trig function follows from those two coordinates: tan θ = y/x, sec θ = 1/x, csc θ = 1/y and cot θ = x/y.

Enter any angle, in degrees or as a radian expression such as 5pi/6, and the calculator marks it on the circle, gives the point in exact radical form when the angle is a multiple of 15°, and reports the quadrant, reference angle, coterminal angles and all six function values. The diagram labels the sixteen standard angles so you can see where yours sits among them.

Any angle outside quadrant I has the same trig values as an acute angle, up to sign. That acute angle is the reference angle, the angle between the terminal ray and the nearest part of the x-axis.

QII: 180° − θ   QIII: θ − 180°   QIV: 360° − θ

The sign comes from the quadrant. In quadrant I all six functions are positive. In quadrant II only sine and cosecant are, in quadrant III only tangent and cotangent, and in quadrant IV only cosine and secant. The mnemonic is All Students Take Calculus, read counterclockwise from quadrant I. The diagram highlights the quadrant label for your angle.

150°

150° is in quadrant II, and its reference angle is 180° − 150° = 30°. The point for 30° is (√3/2, 1/2); in quadrant II x is negative, so the point for 150° is (−√3/2, 1/2). That gives tan 150° = −√3/3, or −0.5774.

7π/4 radians

7π/4 is 315°, in quadrant IV with a reference angle of π/4. The point is (√2/2, −√2/2), so cosine is positive and sine is negative, as the C in ASTC predicts.

A negative angle: −240°

Negative angles turn clockwise. Adding 360° gives the coterminal angle 120°, which lands in quadrant II. The point is (−1/2, √3/2).

An angle with no exact form: 100°

100° is not a multiple of 15°, so there is no simple radical answer. The point is (−0.1736, 0.9848) and the reference angle is 80°. You can still predict the signs: quadrant II, so only sine and cosecant are positive.

270°, on an axis

The point is (0, −1). Because x = 0, tan 270° and sec 270° involve division by zero and are undefined, while cot 270° = 0.

These come from two triangles. A 45-45-90 triangle with hypotenuse 1 has legs √2/2, and a 30-60-90 triangle with hypotenuse 1 has legs 1/2 and √3/2. Reflecting those triangles into each quadrant gives every entry below, generated by the calculator's engine.

A pattern makes them easy to remember. For 0°, 30°, 45°, 60° and 90°, the sine values are √0/2, √1/2, √2/2, √3/2 and √4/2, and the cosine values run the same list backward.

• Exact forms appear only for exact multiples of 15°. An angle like 30.4° gets decimals only, because its sine is not 1/2.
• Radian inputs are read exactly. Typing 5pi/6 is better than 2.618, which is only close to 5π/6 and would lose the exact form.
• Multiples of 15° that are not on the list, such as 15° and 75°, also have exact forms, for example sin 15° = (√6 − √2)/4. The calculator shows them even though the diagram only labels the sixteen common angles.