The unit circle – radius 1, centred at the origin – defines sine and cosine for every angle, not just acute ones. Reading coordinates off it is the fastest way to learn the signs and special values of the trig functions.
How to use the Unit Circle Calculator
- Angle unit
- Angle α
Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.
How it works
Enter the angle and its unit. The calculator places the point, reports its coordinates (exactly when possible), the quadrant, the reference angle and the angle in radians.
Formula
P(α) = (cos α, sin α), x² + y² = 1, tan α = y/x
Worked example
Inputs — Angle unit: Degrees (°); Angle α: 135.
Results — x = cos α = −√2/2 ≈ −0.7071067812; y = sin α = √2/2 ≈ 0.7071067812; tan α = y/x = −1; Quadrant = Quadrant II; Reference angle = 45°; Angle in radians = 2.35619449 rad (3π/4).
Key terms
- Unit circle
- The circle x² + y² = 1.
- Reference angle
- The acute angle to the x-axis.
Frequently asked questions
Why is the radius 1?
So the coordinates are exactly cos α and sin α with no scaling.
What are the coordinates at 45°?
(√2/2, √2/2).
What is a reference angle?
The acute angle between the terminal side and the x-axis.
Do negative angles work?
Yes – they go clockwise from the positive x-axis.