Unit Circle
Drag the point round the circle, or use the arrow keys. Exact values appear at the special angles.
°
0°
sin θ
cos θ
tan θ
Sine and cosine graphs
Arrow keys move 1°, Page Up and Page Down move 15°.
About this tool
The unit circle is a circle of radius 1 centred on the origin. For any angle θ measured anticlockwise from the positive x-axis, the point on the circle is (cos θ, sin θ), and tan θ = sin θ / cos θ. Drag the point and this unit circle shows the angle in degrees and radians, the right-angled triangle under it, the exact values of sin, cos and tan at the special angles (such as √3/2 at 30° and 60°) and the sine and cosine graphs tracing out as you go.
Tips
- CAST tells you which ratio is positive in each quadrant: all in the first, sin in the second, tan in the third, cos in the fourth.
- To turn degrees into radians, multiply by π/180. So 30° = π/6 and 135° = 3π/4.
- tan θ is undefined at 90° and 270° because cos θ is 0 there, and you cannot divide by 0.
FAQ
- Why is the point at (cos θ, sin θ)?
- The radius is the hypotenuse of the right-angled triangle and it has length 1. The horizontal side is 1 × cos θ and the vertical side is 1 × sin θ, so those are the point's coordinates.
- Where do exact values like √3/2 come from?
- From two special triangles. Half of an equilateral triangle with sides 2 gives sides 1, √3 and 2, so sin 60° = √3/2 and cos 60° = 1/2. A right-angled isosceles triangle with legs 1 has hypotenuse √2, so sin 45° = cos 45° = 1/√2 = √2/2.
- What is a reference angle?
- It is the acute angle between the radius and the x-axis. An angle and its reference angle have the same sin, cos and tan apart from the sign, which depends on the quadrant.