How to use the Trig Calculator
- Pick degrees or radians to match how your angle is expressed.
- Enter the angle, including a negative sign for clockwise rotation.
- Read sine, cosine and tangent.
- Note that tangent is reported as undefined at 90° and its odd multiples.
- Use the conversion row to move between the two angle units.
How the calculation works
On the unit circle, an angle measured anticlockwise from the positive x-axis lands on a point whose horizontal coordinate is the cosine and whose vertical coordinate is the sine. Tangent is their quotient, the slope of the radius line. This picture explains everything else: why sine and cosine never leave the range −1 to 1, why the signs flip quadrant by quadrant, and why tangent blows up wherever cosine reaches zero.
Radians are the natural unit and the one every programming language uses internally. One radian is the angle subtending an arc equal to the radius, so a full turn is 2π radians. Degrees are a human convention inherited from Babylonian astronomy. The overwhelming majority of wrong trigonometric answers come from a mode mismatch, which is why this calculator states the unit explicitly and shows the conversion on every result.
Tangent's undefined points are genuine, not a display quirk. As an angle approaches 90°, cosine approaches zero and the quotient grows without bound — the slope of a vertical line. Real applications hit this: a camera pan approaching perpendicular, a projection that tips towards infinity, a numerical solver that diverges near the singularity. Detecting and handling those points is part of using tangent correctly.
sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = sin θ / cos θ; radians = degrees × π/180Source: Unit-circle definitions of the circular functions; see OpenStax Algebra and Trigonometry, Chapter 7.
Worked example
A 30° ramp climbs a fixed 1.2 m rise. How long is the ramp surface and its horizontal run?
- sin(30°) = 0.5, so the ramp length is 1.2 / 0.5 = 2.4 m.
- cos(30°) = 0.8660, so the run is 2.4 × 0.8660 = 2.08 m.
- Check with tangent: tan(30°) = 0.5774 = 1.2 / 2.08. ✓
- In radians, 30° is 0.5236.
A 2.4 m ramp covering a 2.08 m run — far too steep for accessibility, which typically requires closer to 4.8°.
Frequently asked questions
Why is sin(30°) exactly 0.5?+
A 30-60-90 triangle has its shortest side exactly half the hypotenuse, a consequence of bisecting an equilateral triangle.
How do I convert degrees to radians?+
Multiply by π and divide by 180. Ninety degrees is π/2, or about 1.5708 radians.
What are secant, cosecant and cotangent?+
The reciprocals of cosine, sine and tangent respectively. Divide 1 by the displayed value to get them.
Why does tan(90°) sometimes show a huge number elsewhere?+
Because 90° in radians cannot be represented exactly in binary, so cosine lands on a tiny non-zero value instead of true zero. This calculator detects that and reports undefined.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.