Using the Tangent Function to Find the Opposite

Using the Tangent Function to Find the Opposite of a Right Triangle

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

The length of the opposite 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. The image below shows what we mean:

How to Use the Tangent Function to Find the Opposite of a Right Triangle

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

Question

What is the length of the opposite of the right triangle shown below?

Step-by-Step:

1

Start with the formula:

Opposite = tan θ × adjacent

2

Substitute the angle θ and the length of the adjacent into the formula. In our example, θ = 45° and the adjacent is 3 cm.

Opposite = tan (45°) × 3

Adjacent = 1 × 3

Adjacent = 3

Answer:

The length of the opposite of a right triangle with an angle of 45° and an adjacent of 3 cm is 3 cm.

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 one side and one angle and have to find an unknown side...

......think trigonometry...

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

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

Looking at the example above, we are trying to find the Opposite and we know 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

This is rearranged to get the formula at the top of the page (see Note).

Opposite = Tan θ × Adjacent

Slider

The slider below gives another example of finding the opposite of a right triangle (since the angle 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?