Unit Circle Calculator — Trig Values
Enter an angle in degrees or radians to find sin, cos, tan, csc, sec, cot — with exact values for special angles, reference angle, quadrant, and unit-circle coordinates. Converts degrees to radians and back.
How Unit Circle Calculator — Trig Values Works
Find sin, cos, tan, csc, sec, cot values for any angle with exact results, reference angles, and quadrant info. and. Enter your values into the form above and the calculator processes them instantly in your browser — no data is sent to any server.
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 \u03B8, sin \u03B8) where \u03B8 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\u00B0, 30\u00B0, 45\u00B0, 60\u00B0, 90\u00B0 and their equivalents in all four quadrants (120\u00B0, 135\u00B0, 150\u00B0, 180\u00B0, 210\u00B0, 225\u00B0, 240\u00B0, 270\u00B0, 300\u00B0, 315\u00B0, 330\u00B0, 360\u00B0). 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 \u03C0/180. To convert radians to degrees, multiply by 180/\u03C0. For example, 90\u00B0 = 90 \u00D7 \u03C0/180 = \u03C0/2 radians, and \u03C0/3 radians = (\u03C0/3) \u00D7 (180/\u03C0) = 60\u00B0.
What is a reference angle?
A reference angle is the acute angle (between 0\u00B0 and 90\u00B0) 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 \u03B8 = 1/sin \u03B8 (cosecant), sec \u03B8 = 1/cos \u03B8 (secant), and cot \u03B8 = 1/tan \u03B8 (cotangent). They are undefined when their corresponding base function equals zero.