What is the unit circle?

The unit circle is a circle of radius 1 centred at the origin. For any angle measured from the positive x-axis, the point where the angle's ray meets the circle has coordinates equal to the cosine and the sine of that angle. This is why the unit circle underpins all of trigonometry.

How do you find coordinates on the unit circle?

The x-coordinate is the cosine of the angle and the y-coordinate is the sine of the angle. At 30 degrees the point is (cos 30, sin 30) = (0.866, 0.5). The tangent is the y-coordinate divided by the x-coordinate.

What are the 16 standard angles on the unit circle?

The 16 standard angles are 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315 and 330 degrees. These correspond to multiples of 30 and 45 degrees, giving exact values for sine and cosine that are used throughout trigonometry.

Unit Circle Calculator

The unit circle is the single most important picture in trigonometry. It is a circle of radius one centred at the origin, and for any angle measured anticlockwise from the positive x-axis, the point where that angle's ray meets the circle has coordinates that are exactly the cosine and sine of the angle. This calculator takes any angle in degrees or radians and returns the coordinates of that point on the unit circle, the sine, cosine, and tangent values, and the equivalent angle in the other unit. The tangent is the y-coordinate divided by the x-coordinate, which means it is undefined at 90 and 270 degrees where the x-coordinate (cosine) is zero. Understanding the unit circle lets you read off the sign of each trig function in any quadrant without memorising rules separately, because the signs follow directly from whether the x and y coordinates are positive or negative. The table of 16 standard angles below gives you the exact values that appear most often in trigonometry questions: the multiples of 30 degrees (0, 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, 330) and the multiples of 45 degrees (45, 135, 225, 315). These exact values repeat with sign changes in each quadrant, which is why learning one quadrant is enough to know them all. For any angle beyond the standard set, type it into the calculator above and the exact decimal result appears immediately. Use this tool for learning trigonometry, checking values in homework, or as a quick reference during problem-solving.

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Angle

Unit

(0.866, 0.5)

coordinates (cos θ, sin θ) on unit circle

sin θ0.5

cos θ0.866

tan θ0.5774

In radians0.5236

How it works

For an angle θ measured anticlockwise from the positive x-axis, the point on the unit circle is (cos θ, sin θ). The tangent is sin θ divided by cos θ, undefined where cos θ = 0 (at 90 and 270 degrees). Radians convert from degrees by multiplying by π over 180, since a full turn of 360 degrees equals 2π radians.

Worked example

At θ = 30 degrees: cos(30) = √3/2 ≈ 0.866, sin(30) = 1/2 = 0.5. The coordinates on the unit circle are (0.866, 0.5). Tangent = 0.5 / 0.866 ≈ 0.577. In radians, 30 degrees = π/6 ≈ 0.5236. These match the pre-filled defaults above.

16 standard unit circle angles

Degrees	Radians	cos	sin	tan
0	0	1	0	0
30	π/6	0.866	0.5	0.577
45	π/4	0.707	0.707	1
60	π/3	0.5	0.866	1.732
90	π/2	0	1	undef
120	2π/3	-0.5	0.866	-1.732
135	3π/4	-0.707	0.707	-1
150	5π/6	-0.866	0.5	-0.577
180	π	-1	0	0
210	7π/6	-0.866	-0.5	0.577
225	5π/4	-0.707	-0.707	1
240	4π/3	-0.5	-0.866	1.732
270	3π/2	0	-1	undef
300	5π/3	0.5	-0.866	-1.732
315	7π/4	0.707	-0.707	-1
330	11π/6	0.866	-0.5	-0.577

Related calculators

Reference Angle Calculator: find the reference angle and quadrant for 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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