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 The Sine Rule   Sine Rule  or the inverse  Area of triangle    The Cosine Rule Given the triangle ABC as shown

 Useful rules: The largest angle is opposite the largest side, also the smallest angle is opposite the shortest side The two right angled triangles with sides in the ratio 3:4:5 and 5:12:13

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Geometry and Trigonometry

Non-right angled Trigonometry: The Sine Rule and the Cosine Rule

The Sine Rule and the Cosine Rule : Worked Example

Given triangle ABC where AB = 5cm and BC = 7cm and angle C = 40° Find

1. Angle A

2. AC

3. Area of triangle ABC

First sketch the triangle with the information given

(Diagram not to scale)

1. Using the sine rule:

2. To find AC we can use the sine or cosine rule as we now no that angle B = 180 - 40- 64.1 =75.9°

Using the cosine rule:  AC2 = AB2 + BC2 - 2 x AB x BC cos B

Hence   AC2 = 52 + 72 - 2 x 5 x 7 cos 75.9

AC  = 7.55 cm (correct to 3 significant figures)

Using Sine rule

(there is a slight difference in values)

3. Area triangle ABC

 Notes: Try and use values given as much as possible to ensure greater accuracy in answers Do the calculation all in one step on the calculator to avoid rounding errors. Remember brackets on the calculator. Example sin40 should be entered as sin(40)

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