Unit Circle Calculator
Enter an angle to find its coterminal angle, quadrant, reference angle, coordinates, and exact trig values.
Drag & drop or upload an image or PDF
State whether the angle is in degrees or radians. Use pi for π, such as 5pi/4 radians.
Try an example
Coordinates carry the unit-circle values
The unit circle has radius 1 and center at the origin. An angle measured from the positive x-axis meets the circle at (cos θ, sin θ), so the horizontal coordinate is cosine and the vertical coordinate is sine.
For a standard angle, a reference angle gives the exact first-quadrant values and the quadrant supplies the signs. Coterminal angles land at the same point, which means negative angles and angles beyond one full turn use the same process.
How to use the unit circle calculator
Enter the problem as written
Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.
Read the working, not only the answer
Each transformation is separated and explained so you can compare it with your own method.
Ask about any step
Continue in the same solution to request another method, check a restriction, or ask why a rule applies.
The coordinate definition
Every point on the unit circle has coordinates (x, y) with x = cos θ and y = sin θ.
A reliable way to work through it
Reduce to one turn
Add or subtract full turns until the angle lies between 0 and 360 degrees, or between 0 and 2π radians.
Find the reference angle
Measure the acute angle between the terminal side and the x-axis, then use the familiar first-quadrant values.
Apply quadrant signs
Cosine follows the x-coordinate sign, sine follows the y-coordinate sign, and tangent is their ratio.
Worked example
Find the exact unit-circle values for 150°
Its terminal side is above the x-axis and left of the y-axis.
The reference angle is 30 degrees.
Use the exact first-quadrant coordinates for 30 degrees.
Cosine is negative and sine is positive in quadrant II.
Divide the y-coordinate by the x-coordinate and rationalize.
The unit-circle point is (-√3/2, 1/2). Thus sin 150° = 1/2, cos 150° = -√3/2, and tan 150° = -√3/3.
Common mistakes to check
Swapping sine and cosine
A unit-circle point is ordered as (cos θ, sin θ), so cosine is the x-coordinate.
Mixing degrees and radians
A value such as 30 means something entirely different from π/6 unless its unit is stated.
Forgetting quadrant signs
The reference angle gives magnitudes. The actual quadrant determines whether each value is positive or negative.
Calling an undefined ratio infinity
Tangent and secant are undefined when cosine is zero. Cotangent and cosecant are undefined when sine is zero.
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Questions students ask
Why is the unit-circle point written as (cos θ, sin θ)?
On a circle of radius 1, the horizontal coordinate equals cosine and the vertical coordinate equals sine.
Can I enter a negative angle or an angle above 360 degrees?
Yes. Add or subtract full turns to find a coterminal angle at the same point on the circle.
When does the calculator return an exact value?
Standard angles such as multiples of 30 or 45 degrees have exact fractional or radical values. Other angles usually need decimal approximations.
Why is tangent undefined at 90 degrees?
Tangent is sin θ divided by cos θ. At 90 degrees, cosine is zero, so the ratio requires division by zero.