# Using the Tangent Function to Find the Angle

## Using the Tangent Function to Find the Angle of a Right Triangle

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

The angle (labelled θ) is given by the formula below:

In this formula, **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 side next to the angle. tan^{−1} is the inverse tangent function (see **Note**). The image below shows what we mean:

## How to Use the Tangent Function to Find the Angle of a Right Triangle

Finding the angle of a right triangle is easy when we know the opposite and the adjacent.

### Question

What is the angle of the right triangle shown below?

### Step-by-Step:

# 1

Start with the formula:

θ = tan^{−1} (opposite / adjacent)

**Don't forget:** tan^{−1} is the inverse tangent function (it applies to everything in the brackets) **and** / means ÷

# 2

Substitute the length of the opposite and the length of the adjacent into the formula. In our example, the opposite is 5 cm and the adjacent is 5 cm.

θ = tan^{−1} (5 / 5)

θ = tan^{−1} (5 ÷ 5)

θ = tan^{−1} (1)

θ = 45°

### Answer:

The angle of a right triangle with an opposite of 5 cm and an adjacent of 5 cm is 45°.

## Remembering the Formula

Often, the hardest part of finding the unknown angle is remembering which formula to use.

Whenever you have a right triangle where you know two sides and have to find an unknown angle...

......think trigonometry...

...............think sine, cosine or tangent...

........................think **SOH CAH TOA**.

Looking at the example above, we know the **O**pposite and the **A**djacent.

The two letters we are looking for are **OA**, which comes in the **TOA** in SOH CAH **TOA**.

This reminds us of the equation:

**T**an θ = **O**pposite / **A**djacent

We want the angle, θ, not the tangent of the angle, tan θ. To do this, we need to taken the inverse tangent, tan^{−1} (see **Note**).

θ = **T**an^{−1} (**O**pposite / **A**djacent)