Math

Unit Circle Explorer

An interactive unit circle explorer that computes sine, cosine, tangent, cosecant, secant, and cotangent for any angle in degrees or radians. Visualize where the angle falls on the unit circle, identify the quadrant, and find the reference angle instantly.

Unit Circle Tips

Click to show tips

Try an Example

Pick a scenario to see how the calculator works, then adjust the values

30 Degrees

A standard special angle used in trigonometry.

Key values: 30° = π/6 rad · sin = 0.5 · cos ≈ 0.866

45 Degrees

The angle where sine and cosine are equal.

Key values: 45° = π/4 rad · sin ≈ 0.707 · cos ≈ 0.707

90 Degrees

A right angle on the positive y-axis.

Key values: 90° = π/2 rad · sin = 1 · cos = 0

Documentation

Quick Navigation

Jump to any section for specific information about the unit circle and trigonometric functions.

Understanding the Unit Circle

The unit circle is a circle of radius 1 centered at the origin of a coordinate plane. For any angle measured from the positive x-axis, the point where the terminal side of the angle intersects the unit circle has coordinates (cos, sin). This geometric relationship is the foundation of all trigonometric functions.

Every point on the unit circle satisfies the Pythagorean identity:

\cos^2\theta + \sin^2\theta = 1

How to Use This Explorer

Choose your angle unit: Select either degrees or radians.
Enter an angle: Type any angle value. Negative angles and angles beyond 360 degrees (or 2pi radians) are supported and will be normalized.
Read the results: The explorer shows all six trigonometric function values, the coordinates on the unit circle, the quadrant, and the reference angle.

Trigonometric Function Definitions

For a point P = (x, y) on the unit circle at angle theta:

\sin\theta = y \qquad \cos\theta = x

\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}

\csc\theta = \frac{1}{\sin\theta} = \frac{1}{y}

\sec\theta = \frac{1}{\cos\theta} = \frac{1}{x}

\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{x}{y}

Angle Conversion

\text{radians} = \text{degrees} \times \frac{\pi}{180}

Special Angles Reference

These angles have exact trigonometric values that can be expressed without decimals:

Degrees	Radians	sin	cos	tan
0	0	0	1	0
30	pi/6	1/2	sqrt(3)/2	1/sqrt(3)
45	pi/4	sqrt(2)/2	sqrt(2)/2	1
60	pi/3	sqrt(3)/2	1/2	sqrt(3)
90	pi/2	1	0	undefined
180	pi	0	-1	0
270	3pi/2	-1	0	undefined

The ASTC Rule (Sign by Quadrant)

The mnemonic “All Students Take Calculus” helps you remember which trigonometric functions are positive in each quadrant:

Quadrant I (0-90)

All functions are positive

Quadrant II (90-180)

Sine (and csc) are positive

Quadrant III (180-270)

Tangent (and cot) are positive

Quadrant IV (270-360)

Cosine (and sec) are positive

Worked Examples

Example 1: 150 degrees

150 degrees is in Quadrant II. The reference angle is 180 - 150 = 30 degrees.

\sin 150^\circ = \sin 30^\circ = \frac{1}{2}

\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}

Example 2: 5pi/4 radians (225 degrees)

225 degrees is in Quadrant III. The reference angle is 225 - 180 = 45 degrees.

\sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \approx -0.7071

\cos\frac{5\pi}{4} = -\frac{\sqrt{2}}{2} \approx -0.7071

Example 3: -60 degrees

-60 degrees normalizes to 300 degrees, in Quadrant IV. The reference angle is 360 - 300 = 60 degrees.

\sin(-60^\circ) = -\sin 60^\circ = -\frac{\sqrt{3}}{2}

\cos(-60^\circ) = \cos 60^\circ = \frac{1}{2}

Frequently Asked Questions

What is the unit circle?

The unit circle is a circle with radius 1, centered at the origin (0, 0) in the Cartesian coordinate plane. It is the fundamental tool for defining trigonometric functions geometrically.

Why is the unit circle important?

The unit circle provides a visual and geometric way to understand trigonometric functions for all angles, not just those in right triangles. It extends the definitions of sine, cosine, and other trig functions beyond 0 to 90 degrees to all real numbers.

What are special angles on the unit circle?

Special angles are those whose trigonometric values can be expressed as exact fractions or radicals: 0, 30, 45, 60, 90, 120, 135, 150, 180, 210, 225, 240, 270, 300, 315, and 330 degrees. These correspond to multiples and combinations of pi/6, pi/4, and pi/3 radians.

What does it mean when a trig function is undefined?

A trigonometric function is undefined when its computation would involve division by zero. For example, tangent (sin/cos) is undefined when cos = 0, which happens at 90 and 270 degrees. Similarly, cosecant (1/sin) is undefined when sin = 0, at 0 and 180 degrees.

What is the ASTC rule?

ASTC (All Students Take Calculus) is a mnemonic for remembering which trig functions are positive in each quadrant: All functions are positive in Quadrant I, only Sine in Quadrant II, only Tangent in Quadrant III, and only Cosine in Quadrant IV.