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Finding an Angle Using the Sine Rule

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- Introduction

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This worksheet is about finding the length of a side using the sine rule.

Look at triangle ABC below

The sine rule states that:

a | = | b | = | c |

sin A | sin B | sin C |

We can use the sine rule to find an angle in the triangle when we know one other angle and the length of two sides.

*Remember that, if the known angle is between the two given sides, we use the cosine rule.*

__Example:__

In triangle PQR, which is not drawn to scale,

QR = 10 cm, PR = 12 cm and ∠PQR = 95°.

Use the sine rule to find ∠PRQ to 1 dp.

__ Answer__:

First we must find ∠QPR using the sine rule.

Using the sine rule,

12 | = | 10 |

sin 95° | sin ∠QPR |

sin ∠QPR = | 10 x sin 95° |

12 |

sin ∠QPR = 0.83016...

∠QPR = sin^{-1} (0.83016) = 56.1°

**∠PQR** = 180° - 56.1° - 95° = **28.9°**

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