Interactive Unit Circle

The unit circle has radius 1 and its center at the origin. A point at angle θ sits at (cos θ, sin θ). At 45° (π/4 radians) that point is (√2/2, √2/2). Drag the orange point, or type an angle, to see every value change.

How to read the unit circle

For any angle θ measured counterclockwise from the positive x-axis, the point where the terminal side meets the circle is (cos θ, sin θ). The x-coordinate is cosine and the y-coordinate is sine; tan θ = sin θ ÷ cos θ.

The pattern behind the special angles

Only three pairs of numbers appear: (1, 0), (√3/2, 1/2) and (√2/2, √2/2), in some order and with signs. Quadrant I has everything positive; in quadrant II only sine is positive; in III only tangent; in IV only cosine (“All Students Take Calculus”).

Degrees and radians

One full turn is 360° or 2π radians, so multiply degrees by π/180 to get radians: 60° × π/180 = π/3. Radians measure the arc length on the unit circle.

Interactive Unit Circle quiz: 10 questions

Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.

Frequently asked questions

What is the unit circle?

A circle of radius 1 centered at the origin. It defines sine and cosine for every angle as the coordinates of a point on the circle.

How do I memorize the unit circle?

Learn the first quadrant (0°, 30°, 45°, 60°, 90°), then copy it into the other quadrants with the right signs.

Why is it called the unit circle?

Unit means one: its radius is 1, so the coordinates are exactly cos θ and sin θ.

Can I enter angles bigger than 360°?

Yes. The point wraps around: 405° lands in the same place as 45°.

Is there a printable unit circle chart?

Open the chart below the circle and print the page, or see our unit circle article.

Sources

• OpenStax Algebra and Trigonometry 2e: Unit Circle

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