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 Opposite and the Adjacent.

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

This reminds us of the equation:

Tan θ = Opposite / Adjacent

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 = Tan−1 (Opposite / Adjacent)

Slider

The slider below gives another example of finding the angle of a right triangle (if the opposite 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 opposite? Learn more about the tangent function on a right triangle ( interactive widget)