Unit Circle Calculator
Enter any angle in degrees or radians and get the exact sine, cosine, and tangent, plus cotangent, secant, and cosecant, the unit circle coordinates (x, y), the reference angle, and the quadrant. The result updates instantly as you type, with a step-by-step worked solution and a reference table covering all 17 standard angles.
Formula
Worked example
Find all trig values for 150°. Normalized: 150°. Quadrant II (sin+, cos-, tan-). Reference angle = 180° - 150° = 30°. sin(150°) = sin(30°) = 1/2 = 0.5. cos(150°) = -cos(30°) = -root(3)/2 ≈ -0.8660. tan(150°) = 0.5 / (-0.8660) ≈ -0.5774 = -1/root(3). Unit circle point: (-0.8660, 0.5).
What is the unit circle?
The unit circle is a circle with a radius of exactly 1, centered at the origin of a coordinate plane. Because the radius equals 1, every point on the circle has coordinates (x, y) = (cos theta, sin theta), where theta is the angle measured counterclockwise from the positive x-axis. This simple setup turns trigonometry into geometry: you can read the sine and cosine of any angle directly from the y- and x-coordinates of the corresponding point. The unit circle is the foundation of all trigonometry because it defines sine, cosine, and tangent for any angle, not just the acute angles inside a right triangle.
How to use this calculator
Select whether your angle is in degrees or radians, then type the angle. Any real number is valid, including negatives and values beyond 360 degrees - the calculator normalizes the angle automatically. You get sin(theta), cos(theta), tan(theta), plus the reciprocal functions cotangent, secant, and cosecant. The unit circle coordinates (x, y) are shown, along with the reference angle (the acute angle to the nearest x-axis) and the quadrant. The "Show your work" panel walks through the conversion, normalization, reference angle, and each trig value step by step.
Sine, cosine, and all six trig functions
On the unit circle, sin(theta) is the y-coordinate and cos(theta) is the x-coordinate of the point where the terminal side of the angle meets the circle. Tangent is sin divided by cos, which gives the slope of the terminal side from the origin. The three reciprocal functions follow directly: secant = 1 / cos, cosecant = 1 / sin, and cotangent = cos / sin. When cos equals zero (at 90 degrees and 270 degrees) tangent and secant are undefined. When sin equals zero (at 0 degrees and 180 degrees) cotangent and cosecant are undefined. All six values update together whenever you change the angle.
Reference angles and the ASTC sign rule
The reference angle is the smallest positive angle between the terminal side and the nearest x-axis. It always falls between 0 and 90 degrees. Knowing the reference angle lets you find the trig values of any angle from the first-quadrant values alone: the magnitude is the same, but the sign depends on the quadrant. The ASTC rule - All, Sine, Tangent, Cosine - is a memory device: All three primary functions are positive in Quadrant I, only Sine is positive in Quadrant II, only Tangent is positive in Quadrant III, and only Cosine is positive in Quadrant IV. The reference angle in Q2 is 180 degrees minus theta; in Q3 it is theta minus 180 degrees; in Q4 it is 360 degrees minus theta.
Degrees and radians
Degrees and radians are two ways to measure the same rotation. A full circle is 360 degrees or 2*pi radians. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi. The 17 standard angles on the unit circle - multiples of 30 and 45 degrees - have exact values that appear frequently in algebra, calculus, and physics. The reference table below lists all of them with their exact sine, cosine, and tangent values in both degrees and radian notation.
Unit circle values for standard angles
| Degrees | Radians | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -1/√3 |
| 180° | π | 0 | -1 | 0 |
| 210° | 7π/6 | -1/2 | -√3/2 | 1/√3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -1/√3 |
| 360° | 2π | 0 | 1 | 0 |
Exact trig values for the 17 most-used angles. These are the angles worth memorizing for exams.
Frequently asked questions
What is the unit circle used for?
The unit circle defines the sine, cosine, and tangent functions for all real-number angles, not just angles inside a right triangle. It underpins wave equations in physics, Fourier analysis in signal processing, complex number geometry (Euler's formula e^(i*theta) = cos(theta) + i*sin(theta)), and the derivatives and integrals of trig functions in calculus. It is the single most important diagram in trigonometry.
How do I find sine and cosine on the unit circle?
Draw an angle theta from the positive x-axis, measured counterclockwise. Where the terminal side meets the circle is the point (cos theta, sin theta). Because the radius is 1, the horizontal distance from the y-axis to that point equals cos theta, and the vertical distance from the x-axis equals sin theta. You can read both values directly without any calculation.
Why is tan(90°) undefined?
At 90 degrees, cos(90°) = 0. Since tan(theta) = sin(theta) / cos(theta), the formula becomes 1 / 0, which is undefined. Geometrically, the terminal side of a 90-degree angle is vertical: it never intersects a horizontal tangent line, so there is no finite tangent value. The same applies at 270 degrees and any angle that is an odd multiple of 90 degrees.
What are the most important unit circle angles to memorize?
The key angles are 0, 30, 45, 60, and 90 degrees (and their counterparts in the other three quadrants). At these angles the trig values involve only 0, 1, 1/2, root-2/2, and root-3/2. A helpful pattern: sin(0°) = 0, sin(30°) = 1/2, sin(45°) = root(2)/2, sin(60°) = root(3)/2, sin(90°) = 1 - the numerators are root(0), root(1), root(2), root(3), root(4), all divided by 2.
How does the calculator handle angles outside 0 to 360 degrees?
Trig functions are periodic, so any angle gives the same result as its equivalent inside one full revolution. This calculator normalizes the angle by computing angle mod 360 (for degrees) before finding the reference angle and quadrant. A negative angle like -45 degrees is treated as 315 degrees. An angle like 450 degrees maps to 90 degrees. The normalization is shown in the "Show your work" steps whenever it applies.
What is a reference angle?
The reference angle is the acute angle (between 0 and 90 degrees) that the terminal side of your angle makes with the nearest part of the x-axis. It is always positive. In Quadrant I, the reference angle equals the angle itself. In Q2, it is 180 minus the angle. In Q3, it is the angle minus 180. In Q4, it is 360 minus the angle. The sine and cosine of any angle equal, in absolute value, the sine and cosine of its reference angle - only the signs change based on the quadrant.