Trigonometry Calculator — sine, cosine and tangent of an angle

How it works

Trigonometry connects an angle to ratios of side lengths. In a right triangle, for a given angle θ, label the sides opposite it, adjacent to it, and the hypotenuse (the longest, opposite the right angle). The three primary functions are those ratios:

• sin θ = opposite ÷ hypotenuse
• cos θ = adjacent ÷ hypotenuse
• tan θ = opposite ÷ adjacent = sin θ ÷ cos θ

Sine and cosine always fall between −1 and 1. Tangent can be any value and is undefined wherever the cosine is zero — at 90°, 270°, and so on — because you'd be dividing by zero. This calculator takes the angle in degrees.

Worked example

For an angle of 30°:

• sin 30° = 0.5 exactly
• cos 30° = √3 ÷ 2 ≈ 0.8660
• tan 30° = sin ÷ cos ≈ 0.5 ÷ 0.8660 ≈ 0.5774

A few more landmark angles: at 45°, sine and cosine are equal (≈ 0.7071) so tan 45° = 1; at 60°, sin ≈ 0.8660 and cos = 0.5. And at 90°, sin = 1 but cos = 0, so tan 90° is undefined.

Frequently asked questions

What do sine, cosine and tangent mean?

They're ratios of a right triangle's sides for a given angle: sine is the opposite side over the hypotenuse, cosine the adjacent over the hypotenuse, and tangent the opposite over the adjacent. The mnemonic **SOH-CAH-TOA** captures all three. They let you find unknown sides or angles in triangles.

Are the angles in degrees or radians?

This calculator uses **degrees** — the everyday unit where a full turn is 360°. Mathematics often uses radians instead (a full turn is 2π), which the functions convert to internally. If you have radians, multiply by 180 ÷ π to get degrees first.

Why is the tangent undefined at 90°?

Because tan θ = sin θ ÷ cos θ, and cos 90° = 0 — dividing by zero has no value. As the angle approaches 90°, the tangent shoots off toward infinity, so there's no finite answer exactly at 90° (or 270°). The calculator leaves it blank there.

Why do sine and cosine never exceed 1?

Because they're a side divided by the **hypotenuse**, which is always the longest side of a right triangle. A part over the whole longest side can't exceed 1 (or go below −1 once you extend to all angles). Tangent has no such limit, since it compares the two shorter sides.

What are the common exact values?

Worth memorising: sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1 — and cosine is the same list reversed (cos 0° = 1 down to cos 90° = 0). Tangent follows as their ratio: tan 30° = 1/√3, tan 45° = 1, tan 60° = √3.

What is trigonometry used for?

Anywhere angles meet lengths: finding heights and distances you can't measure directly, navigation and surveying, physics (waves, oscillations, projectile motion), engineering, and computer graphics. Sine and cosine also describe anything that cycles — sound, light, alternating current and seasons.