# Tangent Calculator

URL: https://speedycalculator.com/calculators/math/tan/
Last updated: 2026-10-30

> The tangent calculator finds tan α for any angle in degrees, radians or multiples of π, with the exact value for multiples of 15° and the five other trigonometric functions of the same angle.

## Inputs

- Angle unit
- Angle α

## Results

- tan(α)
- sin(α)
- cos(α)
- csc(α)
- sec(α)
- cot(α)

## Formula

```
tan α = opposite / adjacent = sin α / cos α
Range: all real numbers · Domain: α ≠ 90° + k·180° · Period: π (180°)
```

## Explanation

The tangent of an angle is the ratio of the opposite side to the adjacent side – the slope of a line making angle α with the x-axis. It is used for heights and distances, road grades, ramp angles and in navigation.

### How to use the Tangent Calculator

- **Angle unit**
- **Angle α**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

Enter the angle and pick its unit. The angle is converted to radians and tan is evaluated; for multiples of 15° the exact value with square roots is shown as well (for example √3/2). Where the function is undefined, the calculator says so instead of printing a huge number.

## Worked example

Inputs — Angle unit: Degrees (°); Angle α: 30.

Results — **tan(α) = √3/3 ≈ 0.5773502692**; sin(α) = 1/2 ≈ 0.5; cos(α) = √3/2 ≈ 0.8660254038; csc(α) = 2; sec(α) = 1.154700538; cot(α) = 1.732050808.

## Key terms

- **Tangent**: tan α = opposite / adjacent = sin α / cos α.
- **Exact value**: A value written with integers and square roots instead of decimals.

## FAQ

**Why is tan 90° undefined?**
cos 90° = 0, and tangent divides by cosine.

**What is tan 45°?**
Exactly 1 – the opposite and adjacent sides are equal.

**What is the period of tangent?**
180° (π radians), half that of sine and cosine.

**How is tangent related to slope?**
The slope of a line equals the tangent of its angle with the x-axis.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Tangent](https://mathworld.wolfram.com/Tangent.html)