# What is the tangent of my angle?

Finds tan θ of any angle in degrees or radians, with the exact value at multiples of 15°, cot θ, and the reference angle.

- Page: https://www.acalculator.org/math/tangent-calculator
- JSON spec: https://www.acalculator.org/math/tangent-calculator.json
- Version: 5e5b88cd405b

## Default answer

Example with the default inputs (Angle θ 30 °): tan(30°) = 0.57735.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| angle | Angle θ | Any angle, in degrees or radians; a negative angle turns clockwise. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| tan | tan θ | Tangent of θ: sin θ ÷ cos θ, the slope of the line at angle θ. |
| exact | Exact value | The value as a fraction and square roots, for angles that are a multiple of 15°. |
| cot | cot θ | 1 ÷ tan θ; empty where tan θ = 0. |
| reference | Reference angle | The acute angle between θ and the x-axis, from 0° to 90°. |
| quadrant | Quadrant | Where θ ends on the unit circle. |

## Method

tan θ = sin θ ÷ cos θ, the slope of the line from the origin at angle θ. At a multiple of 15° the exact value comes from the special angles and the quadrant sign; otherwise tan θ is computed in 64-bit floating point. cot θ = 1 ÷ tan θ.

## Assumptions

- The angle can be typed in degrees or radians, from −1,000,000 to 1,000,000 radians, and can be negative or more than 360°.
- An angle within one part in 10¹² of a multiple of 15° counts as that multiple, so tan 180° is exactly 0 and tan 90° is undefined.
- tan θ has no answer at 90° + k × 180° (cos θ = 0). cot θ shows nothing where tan θ = 0 (0°, 180°, …).

## Worked examples

1. angle = 0.523599 gives tan = 0.57735, exact = tan 30° = √3/3, cot = 1.732051, quadrant = Quadrant I. Source: OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (tan θ = sin θ ÷ cos θ; tan π/6 = √3/3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions.
2. angle = 0.785398 gives tan = 1, exact = tan 45° = 1, cot = 1. Source: OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (tan θ = sin θ ÷ cos θ; tan π/6 = √3/3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions.
3. angle = 2.094395 gives tan = -1.732051, exact = tan 120° = -√3, quadrant = Quadrant II, reference = 1.047198. Source: OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (tan θ = sin θ ÷ cos θ; tan π/6 = √3/3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions.
4. angle = 1 gives tan = 1.557408, cot = 0.642093. Source: NIST DLMF §4.14 (tangent of a real angle), https://dlmf.nist.gov/4.14.
5. angle = 3.490659 gives tan = 0.36397, quadrant = Quadrant III. Source: NIST DLMF §4.14, https://dlmf.nist.gov/4.14.

## FAQ

### What is the tangent of an angle?

tan θ = sin θ ÷ cos θ. On the unit circle it is y ÷ x for the point at angle θ, the slope of the line from the origin. In a right triangle with acute angle θ, tan θ = opposite side ÷ adjacent side.

### What is tan 45°?

tan 45° = 1 exactly, because sin 45° and cos 45° are both √2/2. Also tan 30° = √3/3 ≈ 0.5774 and tan 60° = √3 ≈ 1.7321.

### Why is tan 90° undefined?

At 90° the point on the unit circle is (0, 1), so tan 90° = 1 ÷ 0, which has no value. The same happens at 270° and at every 90° + k × 180°. The calculator gives no answer there and says why.

### Should I use degrees or radians?

Use the unit the angle is given in. A full turn is 360° or 2π radians. Typing 45 with the unit set to radians gives tan 45 rad ≈ 1.6198, not 1.

### When is tangent negative?

tan θ is negative when sine and cosine have opposite signs: in quadrants II (90° to 180°) and IV (270° to 360°). tan 120° = −√3.

### What is cot θ?

The cotangent, cot θ = 1 ÷ tan θ = cos θ ÷ sin θ. It is 0 where tan θ is undefined, and undefined where tan θ = 0 (0°, 180°), so the calculator leaves it empty there.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (tan t = sin t ÷ cos t; cot t = cos t ÷ sin t; tan π/6 = √3/3). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-01)
- NIST Digital Library of Mathematical Functions, §4.14 Definitions and Periodicity (trigonometric functions). https://dlmf.nist.gov/4.14 (retrieved 2026-10-01)