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

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.

An Example Question

What is the angle of the right triangle shown below?

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 ÷

Step 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°
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)

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

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


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

What Is the Inverse Tangent Function?

The inverse tangent function is the opposite of the tangent function.

The tangent function takes in an angle, and gives the ratio of the opposite to the adjacent:



The inverse tangent function, tan-1, goes the other way. It takes the ratio of the opposite to the adjacent, and gives the angle:

Switch Sides, Invert the Tangent

You may see the tangent function in an equation:



To make theta the subject of the equation, take the inverse tangent of both sides.

The inverse tangent cancels out the tangent on the left hand side of the equals side, so the equation looks as below:



Comparing the two equations, the tangent has moved from one side of the equals sign to the other and has changed from tan to tan-1.

(Note: the reverse is also true. A tan-1 can be moved to the other side of the equals sign, where it becomes a tan.)

Other Inverse Trigonometric Functions

Just as the tangent function has an inverse, so do the sine and cosine functions.