Unit Circle Calculator
Interactive unit circle with trigonometric functions, exact values, and coordinate visualization
Angle Input
Angle Slider
Unit Circle Visualization
Unit Circle Calculator: Exact Trig Values, Radians & Quadrant Reference
The unit circle is a circle of radius 1 centered at the origin. Every point on the circle corresponds to an angle and has coordinates (cosθ, sinθ). Our interactive calculator lets you click any angle to instantly see the exact sin, cos, tan values along with the reference angle and quadrant information.
Special Angles on the Unit Circle — Exact Values
| Angle (°) | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 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 | -√3/3 |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | Undefined |
| 360° | 2π | 0 | 1 | 0 |
How to Use the Unit Circle Calculator
- Enter the angle — type the angle in degrees or radians in the input field, or click directly on the circle visualization.
- Check the mode — confirm whether you are in degrees or radians mode before computing; 90° and π/2 radians are equivalent.
- Read the coordinates — the x-coordinate is the cosine value and the y-coordinate is the sine value for that angle.
- Use the reference table — for special angles (multiples of 30° and 45°), the exact values display as fractions with radicals for precise exam work.
Example Calculations
Example 1 — Find the Exact Value of sin(135°)
- 135° is in Quadrant II — reference angle = 180° − 135° = 45°
- sin(45°) = √2/2 ≈ 0.7071
- In QII, sin is positive → sin(135°) = √2/2
Example 2 — Convert 5π/4 Radians to Degrees and Find cos
- Degrees = (5π/4) × (180/π) = 225°
- 225° is in Quadrant III — reference angle = 225° − 180° = 45°
- cos(45°) = √2/2; in QIII, cos is negative → cos(5π/4) = −√2/2
Frequently Asked Questions
What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin (0, 0) of the coordinate plane. Any point on the circle has coordinates (cosθ, sinθ) where θ is the angle measured counterclockwise from the positive x-axis. This makes the unit circle the foundation of trigonometry — by placing it in a coordinate system, you can define sine, cosine, and tangent for any angle, including angles greater than 90° or negative angles.
How do I memorize the unit circle?
The key trick is memorizing the three "special triangles": 30–60–90 (sides 1, √3, 2) and 45–45–90 (sides 1, 1, √2). From these, derive the six values for 30°, 45°, and 60°. Then use ASTC (All Students Take Calculus) to remember signs by quadrant: QI all positive, QII sine positive, QIII tangent positive, QIV cosine positive. Every other angle on the circle is a reflection of these three reference angles.
What is a reference angle?
A reference angle is the acute angle (0° to 90°) formed between the terminal side of an angle and the x-axis. To find it: QI (0°–090°): reference = θ. QII (90°–180°): reference = 180° − θ. QIII (180°–270°): reference = θ − 180°. QIV (270°–360°): reference = 360° − θ. The trig function value for any angle equals the value for its reference angle, with the sign determined by the quadrant.
Why is tan(90°) undefined?
tan(θ) = sin(θ)/cos(θ). At 90°, cos(90°) = 0, so we would be dividing by zero — which is undefined. On the unit circle, at 90° the point is (0, 1). The tangent function represents the slope of the line from the origin to that point, which is vertical (infinite slope). Similarly, tan(270°) is undefined because cos(270°) = 0. The tangent function has vertical asymptotes at every 90° + 180°n.
How do I convert between degrees and radians?
The conversion is based on the fact that a full circle is 360° = 2π radians. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. Common conversions to memorize: 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π, 30° = π/6, 45° = π/4, 60° = π/3. Radians are preferred in calculus and physics because derivatives of trig functions are cleaner without the degree-to-radian conversion factor.
Related Tools
- Scientific Calculator — Full-featured calc with sin, cos, tan in degrees and radians
- Physics Calculator — Apply trig to kinematics and wave problems
- Statistics Calculator — Statistical analysis and distributions
- Percentage Calculator — Calculate percentage changes and ratios