In this section we use the trigonometric functions to calculate the lengths of sides in a right-angled triangle.

*Trigonometric Functions*

sinΘ = |
cosΘ = |
tanΘ = |

Calculate the length of the side marked *x* in this triangle.

In this question we use the *opposite* side and the *hypotenuse*. These two sides appear in the formula for sin*Θ* , so we begin with,

sin*Θ* =

In this case this gives,

sin 40° | = |

or | |

x | = 8 × sin 40° |

= 5.142300877 cm | |

= 5.1 cm to 1 decimal place |

Calculate the length of the side AB of this triangle.

In this case, we are concerned with side A B which is the *opposite* side and side BC which is the *adjacent* side, so we use the formula,

tan*Θ* =

For this problem we have,

tan 50° | = |

so | |

x | = 9 × tan 50° |

= 10.72578233 cm | |

= 10.7 cm to 1 decimal place |

Calculate the length of the hypotenuse of this triangle.

In this case, we require the formula that links the *adjacent* side and the *hypotenuse*, so we use cos*Θ* .

Starting with

cos*Θ* =

we can use the values from the triangle to obtain,

cos 20° | = |

H × cos 20° | = 12 |

H | = |

= 12.77013327 cm |

Therefore the hypotenuse has length 12.8 cm to 1 decimal place.