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# Finding missing angles in a triangle 0%

Summary

# Finding missing angles in a triangle

## In a nutshell

Calculating the missing angle of a right-angled triangle is very similar to calculating the missing length. However, it is important to first learn about inverse trigonometric functions.

## The inverse trigonometric functions

﻿Inverse trigonometric functions invert the trigonometric function, giving the value of the angle $x$​.

#### examples

$\sin^{-1}()$​​

Inverse sine

If $\sin(x) = 0.5$, then $x = \sin^{-1}(0.5)$​​

$\cos^{-1}()$​​

Inverse cosine

If $\cos(x) = 0.8$​, then $x = \cos^{-1}(0.8)$​​​

$\tan^{-1}()$​​

Inverse tangent

If $\tan(x) = 0.42$​, then $x = \tan^{-1}(0.42)$​​​

Note: The inverse function $\sin^{-1}()$ is NOT the same as $\frac{1}{\sin()}$​! It is just notation to tell us that it is the inverse of the $\sin()$​ function. The same goes for the other two trigonometric functions.

## Calculating the missing angle

The process of finding the missing angle of a right angled triangle is very similar to finding the missing side.

#### PROCEDURE

 1 Label the sides and angle of the triangle. 2 Work out what trigonometric ratio you have to use (SOH, CAH or TOA). 3 Write down the corresponding formula. 4 Substitute in your values for the sides. 5 Perform the appropriate inverse trigonometric function to get the angle.

##### Example

Find the value of $x$ in the diagram to 1 decimal place.

Label the triangle:

You have the opposite and adjacent here. So, you have O and A. Hence, use TOA:

$\tan(x) = \frac{opposite}{adjacent}=\frac{O}{A}$

$\tan(x)=\frac{5}{12}$

Perform the inverse tangent function to get $x$ by itself:

If $\tan(x) = \frac{5}{12}$ then $x = \tan^{-1}(\frac{5}{12})$

$x=22.619864....$​​

$\underline{x=22.6 \degree \ (1 d.p.)}$

## Want to find out more? Check out these other lessons!

Finding missing lengths in a triangle

FAQs

• Question: How do I know when to use the inverse trigonometric functions?

Answer: When you want to reverse a trigonometric function that has the angle inside it. For example, when trying to solve sin(x) = 0.6, use the inverse sine function.

• Question: What do the inverse trigonometric functions do?