Interactive Unit Circle
°
Special angles:
cos θ sin θ tan θ
Drag on the circle to move the angle. It snaps to special angles when you pass near them.
Angle:
Sine, sin θ:
Cosine, cos θ:
Tangent, tan θ:
Point on the circle (cos θ, sin θ):
Where the angle lies:
Reference angle:
Reciprocals:
Calculation:
Move an angle around the unit circle and watch the sine, cosine and tangent change. The cosine is the horizontal segment, the sine is the vertical segment and the tangent is the green segment that starts at the point (1, 0). For special angles the calculator gives the exact value, such as √3/2, as well as the decimal, and tells you the quadrant and the reference angle.
Below you will find the step-by-step method, worked examples, the table of special angles and the most common questions.
How to use the interactive unit circle
Step 1: Drag on the circle, move the slider, tap a special angle or type the angle. Choose Degrees or Radians. In radians you can type pi/6 or 2pi/3.
Step 2: Click Calculate or press Enter if you typed the value. The circle updates instantly.
Step 3: Watch the segments: blue for cos θ, orange for sin θ and green for tan θ. The point on the circle has coordinates (cos θ, sin θ).
Step 4: Read the values, the quadrant and the reference angle in the results. The Calculation box shows how they come out for your angle.
What the unit circle is
The unit circle has radius 1 and its center at the origin. If an angle θ is measured from the positive x-axis, the point of the circle in that direction has coordinates (cos θ, sin θ). That is why cosine is the x-coordinate and sine is the y-coordinate. Because the radius is 1, they always satisfy:
| sin² θ + cos² θ = 1 |
| tan θ = sin θ ÷ cos θ |
The tangent is the green segment because, if you extend the radius to the vertical line x = 1, the height it reaches is tan θ. When cos θ = 0 (90° and 270°) the radius is parallel to that line and the tangent does not exist.
Quadrant and reference angle
The reference angle is the acute angle the radius makes with the x-axis. With it, the values for any angle reduce to those of 0°, 30°, 45°, 60° or 90°, and the quadrant decides the signs.
| Quadrant | Angles | Reference angle | Signs |
|---|---|---|---|
| I | 0° to 90° | θ | sin +, cos +, tan + |
| II | 90° to 180° | 180° - θ | sin +, cos -, tan - |
| III | 180° to 270° | θ - 180° | sin -, cos -, tan + |
| IV | 270° to 360° | 360° - θ | sin -, cos +, tan - |
Unit circle examples
Example 1: 30°
30° = π/6 rad, quadrant I, reference angle 30°
sin 30° = 1/2, cos 30° = √3/2, tan 30° = √3/3
The point on the circle is (√3/2, 1/2) ≈ (0.866, 0.5).
Example 2: 135°
135° = 3π/4 rad, quadrant II, reference angle 180° - 135° = 45°
sin 135° = √2/2, cos 135° = -√2/2, tan 135° = -1
In quadrant II the sine is positive and the cosine is negative, so the tangent comes out negative.
Example 3: 210°
210° = 7π/6 rad, quadrant III, reference angle 210° - 180° = 30°
sin 210° = -1/2, cos 210° = -√3/2, tan 210° = √3/3
In quadrant III the sine and the cosine are both negative, and their quotient, the tangent, is positive.
Example 4: 5π/3 radians
5π/3 × 180° ÷ π = 300°, quadrant IV, reference angle 360° - 300° = 60°
sin = -√3/2, cos = 1/2, tan = -√3
An angle of -60° has the same values, because it is coterminal with 300°: it points to the same place on the circle.
Exact values table for the special angles
| Degrees | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 | -1/2 | -√3 |
| 135° | 3π/4 | √2/2 | -√2/2 | -1 |
| 150° | 5π/6 | 1/2 | -√3/2 | -√3/3 |
| 180° | π | 0 | -1 | 0 |
| 210° | 7π/6 | -1/2 | -√3/2 | √3/3 |
| 225° | 5π/4 | -√2/2 | -√2/2 | 1 |
| 240° | 4π/3 | -√3/2 | -1/2 | √3 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 300° | 5π/3 | -√3/2 | 1/2 | -√3 |
| 315° | 7π/4 | -√2/2 | √2/2 | -1 |
| 330° | 11π/6 | -1/2 | √3/2 | -√3/3 |
| 360° | 2π | 0 | 1 | 0 |
Frequently asked questions
Why is sine the y-coordinate and cosine the x-coordinate?
Because on the unit circle the hypotenuse of the right triangle is 1. The horizontal leg is then cos θ (the hypotenuse times cos θ) and the vertical leg is sin θ. They are the coordinates of the point.
How do I find the sine of 150° without a calculator?
Find the reference angle: 180° - 150° = 30°. The sine of 30° is 1/2, and in quadrant II the sine is positive, so sin 150° = 1/2. The cosine takes a negative sign: cos 150° = -√3/2.
How many degrees is π radians?
π radians is 180°, and 2π radians is a full turn of 360°. To go from degrees to radians, multiply by π and divide by 180. You can check it with the degrees to radians calculator.
Why does the tangent of 90° not exist?
Because tan θ = sin θ ÷ cos θ and cos 90° = 0. You cannot divide by 0. On the circle, the radius at 90° is vertical and never meets the line x = 1, so the green segment cannot be drawn.
What about angles larger than 360° or negative angles?
Each full turn of 360° returns to the same point. 450° points the same way as 90°, and -30° points the same way as 330°. The calculator shows the equivalent angle within one turn.
Learn more
- The unit circle: formulas and diagrams
- Trigonometric ratios of special angles
- Degrees to radians
- Coterminal angles
- Pythagorean identities
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