Unit circle chart, drag an angle, read the values.

A unit circle chart is a circle of radius 1 centred on the origin, used to find sine, cosine and tangent of an angle. Measure the angle from the positive x axis, turning anticlockwise. Where the angle meets the circle, the x coordinate is cos θ and the y coordinate is sin θ. This interactive chart does that for any angle you drag to or type, gives exact values such as √3/2 for every multiple of 15°, and shows the angle in both degrees and radians.

How to use the unit circle chart

1. Drag the purple point round the circle, or click anywhere on it. With Snap to 15° steps when dragging on, it lands on the common angles. Turn it off to move one degree at a time.
2. Or type an angle in Angle and choose Degrees or Radians under Unit. In radians you can write pi or π, so 5π/6 and 5pi/6 both work.
3. Read the cards: the point (cos θ, sin θ), sin θ, cos θ, tan θ, the quadrant, the reference angle, and the angles that land on the same spot.
4. Click Find angles, pick a function and type a value to list every angle from 0° to 360° that gives it.
5. Click Practice for a quiz on the 16 common angles. Copy result copies the line of values and Copy link copies a link that opens on the same angle.

The values to know

Sine of the angles 0°, 30°, 45°, 60° and 90° follows a pattern: √0/2, √1/2, √2/2, √3/2 and √4/2, which simplify to 0, 1/2, √2/2, √3/2 and 1. Cosine is the same list backwards. Every other value on the circle is one of these with a different sign.

Which values are positive in each quadrant

Sine is the height, so it is positive above the x axis. Cosine is the distance across, so it is positive to the right of the y axis. Tangent is sine over cosine, so it is positive when they have the same sign.

Worked examples

• sin 150°. The reference angle is 180° − 150° = 30°. In quadrant II sine is positive, so sin 150° = sin 30° = 1/2. Cosine is negative there, so cos 150° = −√3/2, about −0.866.
• cos 7π/6. Radians to degrees: 7π/6 × 180/π = 210°. The reference angle is 210° − 180° = 30°. Cosine is negative in quadrant III, so the answer is −√3/2.
• tan 225°. The reference angle is 45°, where tan is 1. Tangent is positive in quadrant III, so tan 225° = 1. Check: (−√2/2) ÷ (−√2/2) = 1.
• sin −30°. Add 360° to get 330°, in quadrant IV with reference angle 30°. Sine is negative there, so sin −30° = −1/2.
• sin 405°. Take away one full turn: 405° − 360° = 45°. So sin 405° = √2/2.
• Find the angles where sin θ = √2/2. The calculator finds 45° and 135°. The second is 180° − 45°, because sine is positive in quadrant II too.

Degrees and radians

To change degrees to radians, multiply by π/180. 150° × π/180 = 150π/180 = 5π/6. To go back, multiply by 180/π. A full turn is 360° or 2π radians, and a half turn is 180° or π. The calculator shows the radian form as a multiple of π whenever the angle is a whole number of degrees, and as a decimal otherwise.

Reciprocals and the circle rule

Every point on the circle is 1 away from the centre, so cos² θ + sin² θ = 1. Check it at 60°: (1/2)² + (√3/2)² = 1/4 + 3/4 = 1. The other three functions are reciprocals: csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ. So csc 30° = 2, sec 60° = 2 and cot 45° = 1. Where the bottom of a fraction is 0, the value is undefined, which is why tan 90° has no answer.

Common problems

• Calculator shows a decimal, the answer wants a root. Use the exact card. 0.7071 is √2/2.
• The angle is not a multiple of 15°. No simple exact value exists, so the chart gives decimals rounded to 6 places.
• Mixing up the units. Typing 30 with Radians selected is 30 radians, not 30°. Switch Unit to Degrees first.

For right triangles use the triangle calculator, to graph a line try the slope calculator, and for a full set of functions use the scientific calculator.