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Using the Tangent Function to Find the Adjacent (Mathematics Lesson)

Using the Tangent Function to Find the Adjacent 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 adjacent 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 Adjacent of a Right Triangle

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

An Example Question

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

Step 1
Start with the formula:
Adjacent = opposite / tan θ
Don't forget: / means ÷

Step 2
Substitute the angle θ and the length of the opposite into the formula. In our example, θ = 45° and the opposite is 3 cm.
Adjacent = 3 / tan (45°)
Adjacent = 3 ÷ tan (45°)
Adjacent = 3 ÷ 1
Adjacent = 3
The length of the adjacent of a right triangle with an angle of 45° and an opposite of 3 cm is 3 cm.

Remembering the Formula

Often, the hardest part of finding the unknown adjacent 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 Adjacent and we know the Opposite.



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).
Adjacent = Opposite / Tan θ

A Real Example of How to Use the Tangent Function to Find the Adjacent of a Right Triangle

The slider below gives another example of finding the adjacent of a right triangle (since the angle and opposite are known).
Interactive Widget
Here is an interactive widget to help you learn about the tangent function.
Interactive Test
  show
 


Here's a second test on finding the adjacent using the tangent function.
Here's a third test on finding the adjacent using the tangent function.
Note

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 Adjacent, cover up the A with your thumb:



This leaves O over T - which means O divide by T, or, Opposite ÷ Tan θ.

This tells you that:
Adjacent = Opposite / Tan θ

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.