The Lesson

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

Lesson Slides

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

Interactive Widget

Here is an interactive widget to help you learn about the tangent function on a right triangle.

What Is the Tangent Function?

The tangent function is a trigonometric function. The tangent of a given angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. The tangent function is defined by the formula:

The image below shows what we mean by the given angle (labelled θ), the opposite and the adjacent:

How to Rearrange the Tangent Function Formula

A useful way to remember simple formulae is to use a small triangle, as shown below:

Here, the T stands for Tan θ, the O for Opposite and the A for Adjacent (from the TOA in SOH CAH TOA). To find the formula for the Opposite, cover up the O with your thumb:

This leaves T next to A - which means T times A, or, Tan θ × Opposite. This tells you that:
Opposite = Tan θ × Adjacent

The Tangent Function and the Slope

The slope (or gradient) of a straight line is how steep a line is. It is often defined by "the rise over the run", or how much the line goes up (or down) for how much it goes across.

Looking at the diagram above, the "rise" is the opposite and the "run" is the adjacent. The slope is just tan θ. To find the gradient of a curved line at a certain point, a line is drawn which just touches the curve at that point. This line is called a tangent line, and its slope gives the gradient of the curve at that point.