Trigonometry calculator with exact values for special angles

How to use the trigonometry calculator

1. Choose a Function: sin, cos, tan, csc, sec, cot, or one of the inverse functions sin⁻¹, cos⁻¹, tan⁻¹.
2. Pick the Angle unit: Degrees or Radians.
3. Type the angle in Angle (or ratio for inverse functions). For an inverse function, type the ratio, such as 0.5.
4. Read the Result and, for special angles, the Exact value.
5. Check the All six functions table to see sin, cos, tan, csc, sec and cot of the same angle at once.

What it does and when to use it

Trigonometry links angles to side ratios in right-angled triangles, and to points on the unit circle for any angle. Students use these values to find missing sides and angles, to check exam answers that must be given as surds, and to learn the signs of each function in the four quadrants.

To solve a whole triangle from three known parts, use the triangle solver. For right-angled triangles with two known sides, the Pythagorean calculator is quicker, and the angle calculator finds complementary, supplementary and missing triangle angles.

How it works

Radians are converted with θ(rad) = θ(°) × π ÷ 180. The values come from the standard sine, cosine and tangent functions, and the reciprocals are csc = 1/sin, sec = 1/cos and cot = 1/tan. Results are shown to 12 significant figures.

For whole-degree multiples of 15°, the tool also looks up exact surd forms. It starts from a first-quadrant table (for example sin 15° = (√6 − √2)/4, sin 60° = √3/2) and maps any other angle there using the quadrant rules. If an exact value is 0, the decimal is forced to 0 so you do not see tiny errors like 6.1e-17.

Step by step for cos 135°:

1. 135° lies in the second quadrant (between 90° and 180°).
2. The reference angle is 180° − 135° = 45°.
3. Cosine is negative in the second quadrant.
4. cos 45° = √2/2, so cos 135° = −√2/2 ≈ −0.707106781187.

The tool shows cos(135°) = -0.707106781187 and Exact value: −√2/2.

Worked examples

sin 30°. sin(30°) = 0.5 with Exact value: 1/2. The table also gives cos 30° = √3/2 and tan 30° = √3/3.

tan 90°. The result is undefined, while sin 90° = 1 and cos 90° = 0 in the table.

Inverse cosine. cos⁻¹ with ratio 0.5 gives cos⁻¹(0.5): 60°, which is 1.0471975512 rad.

A non-special angle. sin 37° = 0.601815023152. No exact form exists in surds, so the Exact column shows —.

Limits and tips

• Exact forms are only given for multiples of 15°. Angles like 18° do have surd forms, but the tool shows only the decimal.
• Set the unit before typing. sin(30) in radians is about −0.988, a very different answer.
• All maths runs in your browser.