# Using the Cosine Function to Find the Angle

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

The cosine function relates a given angle to the adjacent side and hypotenuse 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 adjacent is the length of the side next to the angle and the hypotenuse is the length of longest side. cos^{−1} is the inverse cosine function (see **Note**). The image below shows what we mean:

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

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

### Question

What is the angle of the right triangle shown below?

### Step-by-Step:

# 1

Start with the formula:

θ = cos^{−1} (adjacent / hypotenuse)

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

# 2

Substitute the length of the adjacent and the length of the hypotenuse into the formula. In our example, the adjacent is 3 cm and the hypotenuse is 6 cm.

θ = cos^{−1} (3 / 6)

θ = cos^{−1} (3 ÷ 6)

θ = cos^{−1} (0.5)

θ = 60°

### Answer:

The angle of a right triangle with an adjacent of 3 cm and a hypotenuse of 6 cm is 60°.

## 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 **A**djacent and the **H**ypotenuse.

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

This reminds us of the equation:

**C**os θ = **A**djacent / **H**ypotenuse

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

θ = **C**os^{−1} (**A**djacent / **H**ypotenuse)