Unit Circle Calculator — Sin, Cos, Tan & All Trig Functions for Any Angle

What is a unit circle?

A unit circle is a circle with a radius of exactly 1 (unit radius), centered at the origin (0, 0) of the coordinate system. It is one of the most fundamental concepts in trigonometry and forms the basis for defining all six trigonometric functions for any angle — not just acute angles in a right triangle.

Because the radius equals 1, every point on the unit circle satisfies the equation:

x² + y² = 1

Any point A(x, y) on the circumference of the unit circle corresponds to an angle θ measured from the positive x-axis (standard position).

Unit circle: sine and cosine

The power of the unit circle is the simple definition it gives to sine and cosine:

• cos θ = x — the x-coordinate of the point on the unit circle
• sin θ = y — the y-coordinate of the point on the unit circle

This means that for any angle θ, the corresponding point on the unit circle is exactly (cos θ, sin θ). Since the radius is 1, the Pythagorean identity follows directly:

sin²θ + cos²θ = 1

Both sine and cosine always have values between −1 and +1 (inclusive).

Unit circle tangent & other trig functions

All six trigonometric functions are defined on the unit circle:

Unit circle chart — degrees and radians

Angles on the unit circle are measured in both degrees (American/standard system) and radians (metric/SI system). The conversion is:

radians = degrees × π / 180
degrees = radians × 180 / π

The most commonly used angles and their exact values:

Signs in each quadrant

The sign of each trig function depends on which quadrant the angle falls in:

Mnemonic: "All Students Take Calculus" — All (I), Sin (II), Tan (III), Cos (IV) are positive.

Degrees vs Radians: measurement systems

The unit circle naturally works with both measurement systems:

• Degrees (°) — the traditional/American system. A full circle is 360°. Intuitive for everyday use.
• Radians (rad) — the metric/SI standard for angle measurement. A full circle is 2π rad ≈ 6.2832 rad. Radians are preferred in mathematics and physics because they simplify formulas.

Key radian equivalents: π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°, π = 180°, 2π = 360°.

How to memorize the unit circle?

Here are practical strategies to memorize the unit circle values:

1. The "1-2-3 trick" for sin:
   sin(0°) = √0/2 = 0, sin(30°) = √1/2 = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = √4/2 = 1.
   Notice the pattern: 0, 1, 2, 3, 4 under the square root divided by 2!
2. Cos is the reverse:
   cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0.
   Cosine is just sine read backwards from 0° to 90°.
3. Use quadrant symmetry:
   Values in Q2, Q3, Q4 are the same magnitudes as Q1 — just different signs. Apply "All Students Take Calculus" for signs.
4. Remember reference angles:
   Any angle maps to a reference angle in Q1. For 150°, the reference is 30°, so sin(150°) = sin(30°) = 1/2.
5. Practice:
   Use our calculator to check your memorization — enter an angle and verify your expected values.

FAQs

What is the radius of a unit circle?

Exactly 1. That's what makes it a "unit" circle — the radius is one unit.

What is the equation of the unit circle?

x² + y² = 1, centered at the origin (0, 0).

Why is sin θ = y and cos θ = x on the unit circle?

For a right triangle inscribed in the unit circle with hypotenuse = 1, sin = opposite/hypotenuse = y/1 = y, and cos = adjacent/hypotenuse = x/1 = x.

Can angles be greater than 360° or negative?

Yes! Angles greater than 360° simply "wrap around" the circle. Negative angles go clockwise. Our calculator handles any angle value.

When is tan undefined?

When cos θ = 0, i.e., at θ = 90°, 270° (π/2, 3π/2 radians), and their equivalents (±90° + 180°n).

What is the period of sine and cosine?

Both sin and cos have a period of 2π (360°): sin(θ + 2π) = sin(θ).

How do I convert degrees to radians?

Multiply by π/180. Example: 45° × π/180 = π/4 ≈ 0.7854 rad.