Unit Circle Calculator
This unit circle calculator shows the point (cos θ | sin θ) on the circle of radius 1 for any angle. Example: at 150° the point is (−0.866025 | 0.5) and tan θ = −0.57735, in the second quadrant.
Source: Wikipedia – Unit circle. Updated: .
How it is calculated
The unit circle is the circle of radius 1 around the origin. Every angle θ, measured from the positive x-axis counter-clockwise, belongs to a point P(cos θ | sin θ).
How to read the unit circle
The x-coordinate of the point is the cosine, the y-coordinate the sine. The tangent is the ratio sin θ / cos θ, the slope of the segment from the origin to the point. Angles above 360° or below 0° are reduced by full turns. The signs depend on the quadrant: in I sine and cosine are positive, in II cosine is negative, in III both, in IV only sine.
Worked example
For θ = 150° the point lies in the second quadrant; the reference angle is 180° − 150° = 30°. With sin 30° = 1/2 and cos 30° = √3/2, it follows that sin 150° = 1/2 = 0.5 and cos 150° = −√3/2 = −0.866025. The tangent is −√3/3 = −0.57735. The point is therefore P(−0.866025 | 0.5). The calculator also gives the exact values when the angle is a multiple of 30° or 45°.
- Key values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2.
- At 90° the point is (0 | 1); at 180° it is (−1 | 0).
- Radians: 180° corresponds to π ≈ 3.14159 rad.
Frequently asked questions
What does the unit circle show?
For every angle it shows sine and cosine as the coordinates of a point on the circle of radius 1. At 150° that is P(−0.866025 | 0.5).
How do you find the signs of sine and cosine?
From the quadrant: in the second quadrant (90° to 180°) sine is positive and cosine is negative.
Why is the tangent not defined at 90°?
Because cos 90° = 0 and the tangent sin θ / cos θ would need a division by zero.
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