Calculate sine, cosine, tangent, and the (cos, sin) coordinates for any angle on the unit circle, in degrees or radians.
Unit Circle Formula
The unit circle is a circle with a radius of 1 centered at the origin. For any angle theta measured from the positive x-axis, the point where the angle meets the circle has coordinates equal to the cosine and sine of that angle.
(x, y) = (cos theta, sin theta)
The three primary trig functions follow directly from that point:
sin theta = y cos theta = x tan theta = y / x
The three reciprocal functions are:
csc theta = 1 / sin theta sec theta = 1 / cos theta cot theta = cos theta / sin theta
To go the other way and find an angle from a known trig value, you use the inverse functions:
theta = arcsin(value) theta = arccos(value) theta = arctan(value)
- theta: the angle, measured counterclockwise from the positive x-axis, in degrees or radians.
- x: the horizontal coordinate of the point on the circle, equal to cos theta.
- y: the vertical coordinate of the point on the circle, equal to sin theta.
- value: a known function output (between -1 and 1 for sine and cosine) used to solve for the angle.
In the first mode you enter an angle and the calculator returns sin, cos, tan, csc, sec, cot, and the (cos theta, sin theta) coordinates. In the second mode you pick a function and enter its value, and the calculator returns the angles that produce it. Because sine, cosine, and tangent repeat, an inverse value usually has two solutions inside one full turn, so the calculator reports both.
Common Unit Circle Angles
These are the standard angles you are expected to know on the unit circle. The coordinates are exact values.
| Degrees | Radians | cos theta (x) | sin theta (y) | tan theta |
|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 0 |
| 30 | pi/6 | sqrt(3)/2 | 1/2 | sqrt(3)/3 |
| 45 | pi/4 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | 1/2 | sqrt(3)/2 | sqrt(3) |
| 90 | pi/2 | 0 | 1 | undefined |
| 120 | 2pi/3 | -1/2 | sqrt(3)/2 | -sqrt(3) |
| 135 | 3pi/4 | -sqrt(2)/2 | sqrt(2)/2 | -1 |
| 150 | 5pi/6 | -sqrt(3)/2 | 1/2 | -sqrt(3)/3 |
| 180 | pi | -1 | 0 | 0 |
| 270 | 3pi/2 | 0 | -1 | undefined |
| 360 | 2pi | 1 | 0 | 0 |
The sign of each coordinate tells you the quadrant. Cosine (x) is positive on the right half of the circle and sine (y) is positive on the top half.
| Quadrant | Angle range | Positive functions |
|---|---|---|
| I | 0 to 90 | all |
| II | 90 to 180 | sin, csc |
| III | 180 to 270 | tan, cot |
| IV | 270 to 360 | cos, sec |
Example Problems
Example 1: trig values from an angle. Find the unit circle values for theta = 60 degrees. The point is (cos 60, sin 60) = (1/2, sqrt(3)/2), which is about (0.5, 0.866). So sin theta = 0.866, cos theta = 0.5, and tan theta = 0.866 / 0.5 = 1.732. The reciprocals are csc theta = 1.155, sec theta = 2, and cot theta = 0.577.
Example 2: angle from a known value. You know sin theta = 0.5 and want the angles between 0 and 360 degrees. Taking the inverse sine gives theta = 30 degrees. Sine is also positive in the second quadrant, so the other solution is 180 - 30 = 150 degrees. The two answers are 30 degrees and 150 degrees.
FAQ
What is the point of the unit circle? Because the radius is 1, the coordinates of any point on the circle are exactly the cosine and sine of the angle. That makes it a quick reference for the values of trig functions at any angle without doing extra arithmetic.
Why does solving for an angle give two answers? Trig functions repeat as you go around the circle, so more than one angle can produce the same value. For sine and cosine there are normally two angles in a single full turn that share a value, so the calculator reports both. You can add or subtract 360 degrees to find further solutions.
How do I switch between degrees and radians? Use the units selector in the form. One full turn is 360 degrees or 2 pi radians, so to convert degrees to radians you multiply by pi/180, and to convert radians to degrees you multiply by 180/pi.
