Unit Circle with Radians Calculator
Enter an angle in degrees or radians — or tap a radian preset like π/6 or 3π/4 — to find sin, cos, tan, csc, sec, cot with exact values for special angles, the reference angle, quadrant, and unit-circle coordinates.
How the Unit Circle Calculator Works
The unit circle with radians labels every angle by its arc length on a circle of radius 1: a full turn is 2π, a half turn is π, and a quarter turn is π/2. Each point on the circle is (cos θ, sin θ), so π/6 sits at (√3/2, 1/2) and π/4 sits at (√2/2, √2/2).
Enter an angle above (or tap a radian preset) and the calculator returns all six trig values instantly in your browser — no data is sent to any server.
Unit Circle with Radians: Quick-Reference Table
This complete unit circle chart lists all 16 special angles in degrees and radians with their exact coordinates. The x-coordinate is cos θ and the y-coordinate is sin θ. It is the same table the calculator uses for its exact values.
| Degrees | Radians | Point (cos θ, sin θ) |
|---|---|---|
| 0° | 0 | (1, 0) |
| 30° | π/6 | (√3/2, 1/2) |
| 45° | π/4 | (√2/2, √2/2) |
| 60° | π/3 | (1/2, √3/2) |
| 90° | π/2 | (0, 1) |
| 120° | 2π/3 | (−1/2, √3/2) |
| 135° | 3π/4 | (−√2/2, √2/2) |
| 150° | 5π/6 | (−√3/2, 1/2) |
| 180° | π | (−1, 0) |
| 210° | 7π/6 | (−√3/2, −1/2) |
| 225° | 5π/4 | (−√2/2, −√2/2) |
| 240° | 4π/3 | (−1/2, −√3/2) |
| 270° | 3π/2 | (0, −1) |
| 300° | 5π/3 | (1/2, −√3/2) |
| 315° | 7π/4 | (√2/2, −√2/2) |
| 330° | 11π/6 | (√3/2, −1/2) |
Memorizing the unit circle: learn only the first quadrant (0, π/6, π/4, π/3, π/2). Every other angle shares a reference angle with one of these — same denominator, so 5π/6, 7π/6 and 11π/6 all use π/6 — and the CAST rule below gives the signs. To practise for a pre calculus test, fill in a blank unit circle from memory, then check each point with the calculator above.
What Is the Unit Circle?
The unit circle is a circle with radius 1 centered at the origin of the Cartesian plane. Every point on the circle has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. The unit circle provides a geometric way to define all six trigonometric functions for any angle — not just acute angles in a right triangle.
Trig Definitions on the Unit Circle
sin θ = y-coordinate
cos θ = x-coordinate
tan θ = sin θ / cos θ
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = cos θ / sin θ
Conversion: radians = degrees × π / 180
Special Angles
The unit circle has well-known exact values at the "special angles": 0°, 30°, 45°, 60°, and 90° (and their equivalents in all four quadrants). For example:
- sin(30°) = 1/2, cos(30°) = √3/2
- sin(45°) = √2/2, cos(45°) = √2/2
- sin(60°) = √3/2, cos(60°) = 1/2
- sin(90°) = 1, cos(90°) = 0
This calculator recognises all special angles and shows their exact symbolic values alongside decimal approximations.
Reference Angles
A reference angle is the acute angle between the terminal side of your angle and the x-axis. It is always between 0° and 90°. The trigonometric function values for any angle equal those of its reference angle, with the sign determined by the quadrant (CAST rule).
The CAST Rule
CAST tells you which functions are positive in each quadrant. Reading counterclockwise from Q4: Cosine (Q4), All (Q1), Sine (Q2), Tangent (Q3). All other functions in that quadrant are negative. This rule, combined with reference angles, lets you evaluate trig functions without memorizing the entire unit circle.
Pythagorean Identities
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
Applications
The unit circle and trigonometric functions are essential in physics (waves, oscillations, circular motion), engineering (signal processing, AC circuits), computer graphics (rotations, transformations), navigation (bearings, GPS), music (sound waves), and architecture (structural angles). Understanding the unit circle is a prerequisite for calculus, differential equations, and Fourier analysis.
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 can be described as (cos θ, sin θ) where θ is the angle from the positive x-axis. It is the foundation for defining trigonometric functions for all angles.
What are the special angles on the unit circle?
The special angles are 0°, 30°, 45°, 60°, 90° and their equivalents in all four quadrants (120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°). These angles have exact trigonometric values involving fractions, square roots of 2 and 3, and whole numbers.
How do I convert between degrees and radians?
To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. For example, 90° = 90 × π/180 = π/2 radians, and π/3 radians = (π/3) × (180/π) = 60°.
What is a reference angle?
A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of an angle and the x-axis. It is used to find trigonometric values in any quadrant because trig functions of an angle equal the trig functions of its reference angle (with appropriate signs based on the quadrant).
What is the CAST rule?
CAST is a mnemonic for remembering which trigonometric functions are positive in each quadrant. Starting from quadrant IV and going counterclockwise: C (Cosine positive in Q4), A (All positive in Q1), S (Sine positive in Q2), T (Tangent positive in Q3). This helps determine the sign of trig values without memorizing the full unit circle.
What are the reciprocal trig functions?
The reciprocal functions are: csc θ = 1/sin θ (cosecant), sec θ = 1/cos θ (secant), and cot θ = 1/tan θ (cotangent). They are undefined when their corresponding base function equals zero.
What are the radian values on the unit circle?
The 16 special angles in radians are 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π, 7π/6, 5π/4, 4π/3, 3π/2, 5π/3, 7π/4 and 11π/6. One full turn is 2π radians, which equals 360°, so 2π brings you back to the starting point (1, 0).
What is the easiest way to memorize the unit circle?
Learn the first quadrant only: 0 → (1, 0), π/6 → (√3/2, 1/2), π/4 → (√2/2, √2/2), π/3 → (1/2, √3/2) and π/2 → (0, 1). Every other special angle has one of these as its reference angle, so it has the same coordinates with signs set by its quadrant (CAST rule). Practise by filling in a blank unit circle and checking each point with the calculator.