Unit Circle Calculator — Step-by-Step Solutions

The Unit Circle Calculator provides an interactive visualisation of the unit circle with exact trigonometric values for any angle. Click on common angles to see their sine, cosine, and tangent values, or enter any angle to compute all six trig functions.

The calculator supports four modes: an interactive SVG unit circle, angle-to-trig-value lookup with exact values, reference angle calculation, and a degree-radian converter with exact fractions of pi.

Angle (degrees)

Interactive Unit Circle

sin

0.70710678

cos

0.70710678

tan

csc

1.41421356

sec

1.41421356

cot

Common Angles Reference

Deg	sin	cos	tan
0	0	1	0
30	0.5	0.8660254	0.57735027
45	0.70710678	0.70710678	1
60	0.8660254	0.5	1.73205081
90	1	0	undef
120	0.8660254	-0.5	-1.73205081
135	0.70710678	-0.70710678	-1
150	0.5	-0.8660254	-0.57735027
180	0	-1	0
210	-0.5	-0.8660254	0.57735027
225	-0.70710678	-0.70710678	1
240	-0.8660254	-0.5	1.73205081
270	-1	0	undef
300	-0.8660254	0.5	-1.73205081
315	-0.70710678	0.70710678	-1
330	-0.5	0.8660254	-0.57735027

Your calculations will appear here

How to Use the Unit Circle Calculator

Interactive Unit Circle: type an angle in degrees or click one of the marked angles on the circle. The point (cos, sin) is shown on the circle together with its coordinates.
Angle to Trig Values: enter any angle in degrees, including negative angles and angles above 360. You get sin, cos, tan, csc, sec and cot, the exact radian form, and the quadrant. For standard angles such as 30, 45 and 60 degrees, the working shows exact values like sqrt(3)/2.
Reference Angles: enter an angle to see its quadrant, its angle normalised to 0-360 degrees and the acute reference angle, which is what you use to read off the trig values.
Deg / Rad Converter: choose the direction, then enter the value. Degrees to radians gives both a decimal and an exact multiple of pi. Radians to degrees expects a decimal number of radians, so enter pi/3 as 1.0471975512.

Formula and Theory

The unit circle is the circle of radius 1 centred at the origin. For an angle theta measured anticlockwise from the positive x-axis, the point where the angle meets the circle has coordinates (cos theta, sin theta). That is the definition of sine and cosine for any angle, not only acute ones, and it follows that sin^2 + cos^2 = 1.

Tangent is sin / cos, which is the slope of the radius. It is undefined whenever cos = 0, at 90 and 270 degrees. In the same way secant is undefined where cos = 0, and cosecant and cotangent are undefined where sin = 0. The signs follow the quadrant: in quadrant I all functions are positive, in II only sine is, in III only tangent is, and in IV only cosine is.

Every angle has a reference angle, the acute angle between its terminal side and the x-axis. The trig values of an angle equal those of its reference angle up to sign. In quadrant II the reference angle is 180 - theta, in III it is theta - 180, and in IV it is 360 - theta. So you only need to remember the exact values for 30, 45 and 60 degrees.

Radians measure an angle by the arc length it cuts on the unit circle. A full turn is 2 pi radians, so degrees x pi/180 gives radians and radians x 180/pi gives degrees. Calculus and most science formulas assume radians.

Worked Examples

Example 1: Exact values at 30 degrees

Problem: Find sin, cos and tan of 30 degrees and write the angle in radians.

Solution: The angle is in quadrant I, so it is its own reference angle. From the 30-60-90 triangle: sin 30 = 1/2, cos 30 = sqrt(3)/2 = 0.8660 and tan 30 = sin/cos = (1/2) / (sqrt(3)/2) = 1/sqrt(3) = 0.5774. In radians, 30 x pi/180 = pi/6.

Answer: sin = 1/2, cos = sqrt(3)/2, tan = 1/sqrt(3); 30 degrees = pi/6

Example 2: An angle in the second quadrant

Problem: Find sin, cos and tan of 135 degrees.

Solution: 135 degrees lies in quadrant II, so the reference angle is 180 - 135 = 45 degrees. For 45 degrees, sin = cos = sqrt(2)/2. In quadrant II sine is positive and cosine negative, so sin 135 = sqrt(2)/2 = 0.7071, cos 135 = -sqrt(2)/2 = -0.7071 and tan 135 = -1.

Answer: sin = sqrt(2)/2, cos = -sqrt(2)/2, tan = -1; reference angle 45 degrees

Example 3: An angle larger than 360 degrees

Problem: Find the trig values of 480 degrees.

Solution: Subtract one full turn: 480 - 360 = 120 degrees, which is in quadrant II. The reference angle is 180 - 120 = 60 degrees. With sin 60 = sqrt(3)/2 and cos 60 = 1/2, signs for quadrant II give sin 480 = sqrt(3)/2 = 0.8660, cos 480 = -1/2 and tan 480 = -sqrt(3) = -1.7321.

Answer: sin = sqrt(3)/2, cos = -1/2, tan = -sqrt(3)

Example 4: Reference angle in the third quadrant

Problem: What is the reference angle of 225 degrees?

Solution: 225 degrees lies between 180 and 270, so it is in quadrant III. The reference angle is 225 - 180 = 45 degrees.

Answer: 45 degrees (quadrant III)

Example 5: Converting between degrees and radians

Problem: Convert 150 degrees to radians and 2 radians to degrees.

Solution: 150 x pi/180 = 5 pi / 6, which is 2.6180 radians. For the reverse, 2 x 180/pi = 114.5916 degrees.

Answer: 150 degrees = 5pi/6 = 2.6180 rad; 2 rad = 114.59 degrees

Frequently Asked Questions

How do I remember the unit circle?

Learn the first quadrant. The sines of 0, 30, 45, 60 and 90 degrees are sqrt(0)/2, sqrt(1)/2, sqrt(2)/2, sqrt(3)/2 and sqrt(4)/2, and the cosines are the same list in reverse. The other quadrants only change the signs, using the reference angle.

Why is tan 90 degrees undefined?

Because tan = sin / cos and cos 90 = 0. Division by zero is undefined, and the tangent graph has a vertical asymptote at 90 degrees. The same happens at 270 degrees.

What is a reference angle?

It is the acute angle between the terminal side of the angle and the x-axis, always between 0 and 90 degrees. It lets you use the known values for 30, 45 and 60 degrees at any angle.

How do negative angles work?

A negative angle is measured clockwise. -45 degrees ends at the same place as 315 degrees, so sin(-45) = -sqrt(2)/2 and cos(-45) = sqrt(2)/2. The calculator normalises angles to the range 0 to 360 for you.

Should my calculator be in degrees or radians?

It depends on the problem. This tool takes degrees for angles, and the converter changes between the two. A scientific calculator left in the wrong mode is the most common source of wrong trig answers.

What are the signs of sine and cosine in each quadrant?

Quadrant I: both positive. II: sine positive, cosine negative. III: both negative. IV: sine negative, cosine positive. The phrase "All Students Take Calculus" lists which functions are positive in I to IV.