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 Adjacent and the Hypotenuse.

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

This reminds us of the equation:

Cos θ = Adjacent / Hypotenuse

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 = Cos−1 (Adjacent / Hypotenuse)

Slider

The slider below gives another example of finding the angle of a right triangle (if the hypotenuse and adjacent are known).

Open the slider in a new tab

See Also

What is an angle? What is a right triangle? What is the adjacent? What is the hypotenuse?