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Unit Circle Angle Finder

Find sine, cosine and tangent for any angle on the unit circle.

What is the Unit Circle Angle Finder?

A unit circle angle finder calculates the sine, cosine and tangent of any angle, entered in either degrees or radians. It works by converting degrees to radians where needed (radians = degrees × π ÷ 180), then applying the standard trigonometric functions to that angle — sine and cosine come directly from the y and x coordinates of a point on a circle of radius 1, while tangent is sine divided by cosine (which becomes undefined where cosine is zero, at 90° and 270°). To calculate it by hand for common angles, memorise the unit circle values for 0°, 30°, 45°, 60° and 90° and use symmetry to find values for other angles around the circle, or use a scientific calculator set to the correct angle mode (degrees or radians).

How it works

On a unit circle (a circle with radius exactly 1, centred at the origin), any angle measured from the positive x-axis corresponds to a specific point on the circle — the x-coordinate of that point is the cosine of the angle, and the y-coordinate is the sine. Tangent is calculated as sine divided by cosine, which is undefined at angles where cosine equals zero (90° and 270°, and their equivalents).

UK context

Working in radians rather than degrees is standard in most higher mathematics and programming contexts, since radians relate angle directly to arc length on the unit circle (a full circle is 2π radians) — but degrees remain the more intuitive everyday unit, which is why being able to convert and calculate in either is useful.

Tips

Remember that tangent is undefined at 90° and 270° (and any angle that's an odd multiple of 90°), since cosine is zero at those points and division by zero is undefined.
Sine and cosine values always fall between -1 and 1 inclusive, since they're coordinates on a circle of radius 1 — a result outside that range signals a calculation error somewhere.

Frequently asked questions

What's the relationship between sine, cosine and the unit circle?

For any angle measured from the positive x-axis on a unit circle, cosine gives the x-coordinate of the corresponding point and sine gives the y-coordinate — this is the geometric definition behind the trigonometric functions.

Why is tangent sometimes undefined?

Tangent is calculated as sine divided by cosine, so it's undefined wherever cosine equals zero — this happens at 90° and 270° (and their equivalents), where the unit circle point sits directly on the y-axis.