# Trigonometry: Finding angles and sides

## In a nutshell

$\sin()$, $\cos()$, $\tan()$ and their corresponding ratios can be used to find missing lengths and angles in a right-angled triangle.

## Calculating a 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 and rearrange the formula to make the missing side the subject. |

4. | Substitute in the values for the other side and the angle to get the length. |

##### Example 1

*Find the value of the side labelled *$x$* in the diagram given to 1 decimal place.*

*Label the sides and angle:*

$O$* and *$H$* are the given sides. So, use SOH:*

$\sin(50) = \frac{O}{H}$

*So:*

$O = H \times \sin(50)$

*Substitute in the values*:

$O = 10 \times \sin(50) = 7.660444....$

*Round to 1 decimal place, like the question asks:*

$\underline{O = 7.7m \ (1 \ d.p.)}$

## Calculating an angle

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 the values for the sides. |

5. | Perform the appropriate inverse trigonometric function to get the angle. |

##### Example 2

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

*First, label the diagram:*

*$O$ and $A$ are the important sides here. Hence, use TOA:*

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

*Substitute in the values:*

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

*Use the corresponding inverse trigonometric function:*

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

*$x= 22.61986....$*

*Round to 1 decimal place, like the question asks:*

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