Unit Circle Calculator

Enter any angle in degrees or radians and the Unit Circle Calculator returns the exact coordinates (x, y) on the unit circle, along with sin(α), cos(α), tan(α), and the other three reciprocal trig functions. Choose your preferred angle mode, type in a value like 45° or π/4, and see all six trig values plus a visual breakdown of the unit circle point.

Enter the angle in the selected unit. Any value is accepted — the calculator maps it to the unit circle.

Results

Coordinates (x, y)

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sin(α)

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cos(α)

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tan(α)

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csc(α)

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sec(α)

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cot(α)

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Angle in Degrees

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Angle in Radians

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sin(α) vs cos(α) Proportion

Results Table

Frequently Asked Questions

What is a unit circle?

A unit circle is a circle with a radius of exactly 1, typically centered at the origin (0, 0) of a coordinate plane. It is a foundational concept in trigonometry because the x and y coordinates of any point on the circle directly equal the cosine and sine of the corresponding angle, respectively.

How do sin and cos relate to the unit circle?

For any angle α measured from the positive x-axis, the point on the unit circle is (cos α, sin α). Because the radius is 1, the hypotenuse is always 1, making sin(α) = y/1 = y and cos(α) = x/1 = x. This geometric interpretation is why the unit circle is so powerful in trigonometry.

How do I find tan using the unit circle?

Tangent is defined as sin(α) / cos(α), which on the unit circle equals y / x. For example, at 30° sin(30°) = 0.5 and cos(30°) ≈ 0.866, so tan(30°) ≈ 0.577. Tangent is undefined whenever cos(α) = 0, i.e., at 90° and 270°.

What is tan 30° using the unit circle?

At 30°, sin(30°) = 1/2 and cos(30°) = √3/2. Dividing gives tan(30°) = (1/2) / (√3/2) = 1/√3 = √3/3 ≈ 0.5774. This is one of the standard exact values found on the unit circle chart.

How do I find cosecant (csc) with the unit circle?

Cosecant is the reciprocal of sine: csc(α) = 1 / sin(α). On the unit circle, sin(α) = y, so csc(α) = 1/y. It is undefined when y = 0, i.e., at 0° and 180°. Simply calculate sin first and take its reciprocal.

How do I find arcsin(1/2) with the unit circle?

arcsin(1/2) asks: for which angle is sin(α) = 1/2? Looking at the unit circle, y = 1/2 at α = 30° (π/6) in the first quadrant and α = 150° (5π/6) in the second quadrant. The principal value (output of arcsin) is 30° or π/6.

How can I memorize the unit circle?

A common trick is to remember the sine values for the key angles 0°, 30°, 45°, 60°, 90° as √0/2, √1/2, √2/2, √3/2, √4/2 — which simplify to 0, 0.5, ≈0.707, ≈0.866, 1. Cosine values follow the reverse pattern. Practicing with a unit circle chart and associating angles with their (x, y) coordinates helps reinforce memory.

Does the calculator work with angles greater than 360° or negative angles?

Yes. The unit circle repeats every 360° (or 2π radians), so the calculator automatically maps any angle — including large positive values or negative angles — to its equivalent position on the circle. For example, 390° gives the same result as 30°, and −45° corresponds to 315°.

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