# What Is the Tangent Function?

## What Is the Tangent Function?

The tangent function relates a given angle to the opposite side and adjacent side of a right triangle.

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.

### Dictionary Definition

The Merriam-Webster dictionary defines the tangent function as "the trigonometric function that for an acute angle is the ratio between the leg opposite to the angle when it is considered part of a right triangle and the leg adjacent."

The tangent of an angle is given by the formula below:

In this formula, **tan** denotes the tangent function, **θ** is an angle of a right triangle, the opposite is the length of the side opposite the angle and the adjacent is the length of the side adjacent the angle. The image below shows what we mean:

## Interactive Widget

Use this **interactive widget** to create a right-angled triangle and then use the tangent function to calculate the hidden element. Start by selecting which element you want to hide (using the green buttons) and then clicking in the shaded area.

Angle: 0° | |

Opposite: ? | |

Adjacent: 0 |

**Oops, it's broken!Turn your phone on its side to use this widget.**

## A Real Example of the Tangent Function

It is easier to understand the tangent function with an example.

### Question

Find tan 45° using the right triangle shown below.

### Step-by-Step:

# 1

Start with the formula:

tan θ = opposite / adjacent

**Don't forget:** / means ÷

# 2

Substitute the angle θ, the length of the opposite and the length of the adjacent into the formula. In our example, θ = 45°, the opposite is 4 cm and the adjacent is 4 cm.

tan (45°) = 4 / 4

tan (45°) = 4 ÷ 4

tan (45°) = 1

### Answer:

tan 45° = 1.

## The Graph of the Tangent Function

The tangent function can be plotted on a graph.

The tangent function is not defined at 90°, 270° (and any amount of 180° added or subtracted from these angles). This is seen on the graph as vertical red dashed lines, which the tangent function approaches but never touches. These lines are called *asymptotes* (see **Note**.)

Find the angle along the horizontal axis, then go up until you reach the tangent graph. Go across and read the value of tan θ from the vertical axis.

We can see from the graph above that tan 45° = 1.

## The Tangent Function and the Unit Circle

The tangent function can be related to a unit circle, which is a circle with a radius of 1 that is centered at the origin in the Cartesian coordinate system.

For a point at any angle θ, tan θ is given by the ratio of the y-coordinate to the x-coordinate of the point.