In this section we introduce 3 functions: *sine*, *cosine* and *tangent*, and their use in right-angled triangles. First we look at the conventions used for the names of the sides of a right-angled triangle with respect to one of the angles.

The *adjacent* side is the side joining the angle and the right angle.

The *opposite* side is opposite the angle.

The *hypotenuse* is the side opposite the right angle and is the longest side in the triangle.

Using these definitions, we can write down the trigonometric functions:

sinΘ = | = | |

cosΘ = | = | |

tanΘ = | = |

Note that we abbreviate sine, cosine and tangent to sin, cos and tan.

In the following Examples and Exercises, we investigate the properties of these trigonometric functions.

Estimate the sin, cos and tan of 30°, using an accurate drawing of the triangle shown.

The triangle has been drawn accurately below, and the sides measured.

Here, hypotenuse = 11.6 cm , adjacent = 10 cm and opposite = 5.8 cm, so,

sin30° | = = | 0.5 |

cos30° | = = | 0.86 (to 2 d.p.) |

tan30° | = = | 0.58 |

Note that if we had drawn a similar right-angled triangle, again containing the 30° angle but with different side lengths, then we may have obtained slightly different values for sin30°, cos30° and tan30°. You can obtain more accurate values of sin30°, cos30° and tan30° by using a scientific calculator. If you have a calculator with the trigonometric functions, do this and compare them with the values above.

WARNING: When you use a scientific calculator, always check that it is dealing with angles* in degree mode*.

(a)

Measure the angle marked in the following triangle:

In this case the angle can be measured with a protractor as 37°.

(b)

Calculate the sine, cosine and tangent of this angle.

Here we have

opposite | = 6 cm |

adjacent | = 8 cm |

hypotenuse | = 10 cm |

sinΘ | = | = | = 0.6 |

cosΘ | = | = | = 0.8 |

tanΘ | = | = | = 0.75 |