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Using the Sine or Cosine Rule

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This worksheet is about using the sine and the cosine rule to find the lengths of sides or the sizes of angles.

Look at triangle ABC below

The cosine rule states that:

a^{2} = b^{2} + c^{2} - 2bc cosA

We can use the cosine rule to find an angle when we know all three sides of a triangle.

We can use the cosine rule to find the length of a side when we know two sides of a triangle and the angle between these two sides.

__Example 1__

In triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and PR = 15 cm.

Use the cosine rule to find ∠PQR to the nearest degree.

__ Answer__:

Using the cosine rule,

15^{2} = 9^{2} + 10^{2} - 2 x 9 x 10 x cosθº

225 = 81 + 100 - 180cosθº

180cosθº = 81 + 100 - 225 = -44

cosθº = -44 ÷ 180 = -0.24444444....

**θº** = cos^{-1}(-0.24444444...) = **104°
**

__Example 2__

In triangle PQR, which is not drawn to scale, PQ = 7 cm, QR = 22 cm, and ∠PQR = 145º.

Find the length PR to 3 sig figs.

__ Answer__:

Using the cosine rule,

PR^{2} = 7^{2} + 22^{2} - 2 x 7 x 22 x cos145º

= 49 + 484 - 308 x (-0.819)

= 785.3

**PR** = √785.3 ≈ **28.0 cm**

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.*

We can use the sine rule to find the length of a side when we know two angles and an opposite side.

__Example 3__

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°**

__Example 4__

In triangle PQR, which is not drawn to scale,

PQ = 7 cm, ∠QPR = 72° and ∠PRQ = 24°.

Use the sine rule to find the length of PR in cm to 1 dp.

__ Answer__:

First we must find ∠PQR = 180° - 72° - 24° = 84°.

Using the sine rule,

PR | = | 7 |

sin 84° | sin 24° |

PR = | 7 x sin 84° |

sin 24° |

PR = 7 x sin 84° ÷ sin 24°

**PR = 17.1 cm**

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