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Unit Circle

An interactive unit circle with angles, radians, sine and cosine.

0°30°45°60°90°120°135°150°180°210°225°240°270°300°315°330°

Drag the point, tap an angle, or use the slider.

=

cos θ
sin θ
tan θ

The point on the circle is , which is (cos θ, sin θ).

Special angles

DegreesRadianscossintan

When to use it

Use this interactive unit circle to see how angles relate to sine, cosine and tangent. It helps students learning trigonometry for the first time, revising for exams, or checking exact values, and teachers showing how the values change as an angle moves around the circle.

How to use it

  1. Drag the point around the circle, tap an angle, or use the Angle slider.
  2. Or type an exact angle in Angle in degrees.
  3. Read the angle in Degrees and Radians, and the values of cos θ, sin θ and tan θ.
  4. Tick Snap to special angles to jump to angles like 30°, 45° and 60° while dragging.
  5. Tick Show tangent line to see tan θ drawn on the diagram.
  6. Use the Special angles table as a quick reference.

The sentence under the diagram tells you the point’s coordinates and which quadrant it is in.

Worked example

Move the point to 60°. The tool shows π/3 radians, cos θ = 1/2, sin θ = √3/2 and tan θ = √3. The point on the circle is (0.5, 0.866), which is (cos θ, sin θ).

Now drag to 120°, in quadrant II. The sine stays √3/2, but the cosine becomes −1/2, and tan becomes −√3. Seeing the point move to the left of the y-axis makes it clear why cos turned negative.

Patterns to notice

  • sin and cos swap between 30° and 60°: cos 30° = sin 60° = √3/2.
  • At 45°, sin and cos are equal, so tan 45° = 1.
  • Angles 180° apart have the same tan value.
  • After 360°, the values repeat.

Why it matters

The unit circle is the key to graphs of sine and cosine, to waves in physics, and to rotation in computer graphics. If you can picture the point moving around the circle, the up-and-down shape of a sine wave makes sense: it is just the height of that point over time.

Tips

  • Learn the first quadrant well. Every other value is the same number with a different sign.
  • Use snap for exact values. It avoids landing on 59° when you meant 60°.
  • Check your calculator mode. If your answers disagree with the tool, your calculator may be set to radians.

Common mistakes

  • Swapping sin and cos. Remember: x is cos, y is sin, in alphabetical order.
  • Forgetting negative signs. Check the quadrant before writing an answer.
  • Mixing degrees and radians. π/3 and 60° are the same angle, but 60 radians is not.

Frequently asked questions

What is the unit circle?

A circle with a radius of 1, centred at the origin. For any angle θ measured from the positive x-axis, the point where the angle meets the circle has coordinates (cos θ, sin θ). That simple fact links trigonometry to geometry.

Why is tan undefined at 90° and 270°?

tan θ is sin θ divided by cos θ. At 90° and 270°, cos θ is 0, and you cannot divide by zero, so tan has no value there.

How do I convert degrees to radians?

Multiply by π and divide by 180. So 60° is 60π/180 = π/3. A full turn of 360° is 2π radians. The tool shows both for every angle.

What are the special angles?

The multiples of 30° and 45°, such as 30°, 45°, 60° and 90°, and their matches in the other quadrants. Their sine and cosine have exact values with square roots, like √3/2, which you are often expected to know.

Why do signs change in different quadrants?

Cosine is the x-coordinate and sine is the y-coordinate. In quadrant II, x is negative, so cos is negative. In quadrant III both are negative. In quadrant IV, y is negative. A common memory aid is "All Students Take Calculus" for which values are positive in quadrants I to IV.

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