Interactive Unit Circle

The unit circle is a circle of radius 1 centered at the origin. Start at the point (1, 0) on the positive x-axis and turn counterclockwise by an angle θ. Where you stop is the point (cos θ, sin θ): the x-coordinate is the cosine and the y-coordinate is the sine. Turning clockwise gives a negative angle, so -45° lands on the same point as 315°.

Because the radius is 1, the point, the x-axis and a vertical drop from the point form a right triangle with hypotenuse 1. In the drawing above the blue leg along the x-axis has signed length cos θ and the red vertical leg has signed length sin θ. That is why x² + y² = 1 becomes the identity sin²θ + cos²θ = 1. The other four functions come from these two: tan θ = y/x, cot θ = x/y, sec θ = 1/x and csc θ = 1/y, so any of them is not defined when its denominator is 0. For example, tan 90° = 1/0 is not defined, while cot 90° = 0/1 = 0.

These definitions follow OpenStax Precalculus 2e, sections 5.2 and 5.3, which define sine and cosine as the coordinates of the terminal point and build the other four functions from them.

Every exact value on the circle comes from two special right triangles scaled to a hypotenuse of 1.

• 45-45-90: both legs are equal, so x = y and 2x² = 1, which gives x = y = √2/2. The 45° point is (√2/2, √2/2).
• 30-60-90: the short leg is half the hypotenuse, 1/2, and the long leg is √3/2. At 30° the long leg lies along the x-axis, giving (√3/2, 1/2). At 60° the triangle stands up and the values swap to (1/2, √3/2).

Every other standard angle reuses one of these triangles through its reference angle, the acute angle between the terminal side and the x-axis. For 150° the reference angle is 180° - 150° = 30°, for 210° it is 210° - 180° = 30°, and for 300° it is 360° - 300° = 60°. Take the values for the reference angle and attach the signs of the quadrant: 210° is (-√3/2, -1/2).

Here is the same sequence of steps the tool follows when you enter 240°.

1. Normalize: 240° is already between 0° and 360°.
2. Quadrant: 240° is between 180° and 270°, so it is in Quadrant III, where x and y are both negative.
3. Reference angle: 240° - 180° = 60°.
4. Coordinates: the 60° point is (1/2, √3/2). Attaching Quadrant III signs gives (-1/2, -√3/2), so cos 240° = -1/2 and sin 240° = -√3/2 ≈ -0.866.
5. Tangent and cotangent: tan = y/x = (-√3/2)/(-1/2) = √3 ≈ 1.7321, and cot = x/y = 1/√3 = √3/3 ≈ 0.5774. Both are positive, as ASTC predicts for Quadrant III.
6. Reciprocals: sec = 1/x = -2 and csc = 1/y = -2/√3 = -2√3/3 ≈ -1.1547.
7. Radians: 240 × π/180 = 4π/3 ≈ 4.1888.

Decimal values in the tool are rounded to 4 places. For an angle that is not one of the 16 standard angles, such as 20°, there is no simple radical form, so only the decimal is shown: sin 20° ≈ 0.342.

A radian is the angle that cuts off an arc equal to the radius. On the unit circle the radius is 1, so the radian measure of an angle is simply the length of the arc it sweeps. A full turn is 2π radians, which gives the conversions radians = degrees × π/180 and degrees = radians × 180/π.

Standard angles are easiest to remember as fractions of π. Multiples of 30° are sixths (30° = π/6, 150° = 5π/6, 330° = 11π/6), multiples of 45° are fourths (135° = 3π/4, 225° = 5π/4), multiples of 60° are thirds (120° = 2π/3, 240° = 4π/3), and the axes are 0, π/2, π and 3π/2. As a decimal, 5π/6 is about 2.618. You can type either form in the angle box with the unit set to radians, for example 5π/6, 5pi/6 or 2.618.

Angles past a full turn or below zero are coterminal with an angle between 0° and 360°: add or subtract 360° (or 2π) until it is in range. 480° and -240° both land on 120°, so they share its sine, cosine and tangent.