In this section we use trigonometry to determine the sizes of angles in rightangled triangles. On your scientific calculator you will find buttons labelled 'sin^{–1}' , 'cos^{–1}' and 'tan^{–1}' . You will need to be able to use these to calculate the angles that will arise in the problems which follow. Again, we start with the three trigonometric functions:

sinΘ = |
cosΘ = |
tanΘ = |

Calculate the angle *Θ* in this triangle.

In this triangle we are given the lengths of the adjacent and opposite sides, so we will use,

tan*Θ* =

Using the lengths given, we have

tanΘ | = |

= 1.6 |

We can then use the tan^{–1} key on a calculator to obtain

Θ | = tan^{–1}(1.6) = 57.99461678° |

= 58.0° (to 1 decimal place) |

Calculate the angle marked *Θ* in this triangle.

Because the lengths given are for the adjacent side and the hypotenuse, the formula for cos*Θ* must be used.

cos Θ | = |

= = 0.470588235 | |

Θ | = cos^{–1}(0.470588235) = 61.92751306° |

= 61.9° (to 1 decimal place) |

A rectangle has sides of length 5 m and 10 m. Determine the angle between the long side of the rectangle and a diagonal.

The solution is illustrated in the diagram.

Using the formula for tan*Θ* gives

tan Θ | = |

= 0.5 |

Then using the tan^{–1} key on a calculator gives

Θ | = tan^{–1}(0.5) = 26.56505118° |

= 26.6° (to 1 decimal place) |