# 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 $x$.

#### Function | #### What it's called | #### 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:*

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

*Substitute in your values:*

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

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

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

**

$x=22.619864....$

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