Download The Cosine Rule - National 5 Maths

The Cosine Rule
Theory in a Nutshell
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Used to calculate angles & lengths in non-right angled triangles
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Formula given in National 5 exam
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If calculating a length (a) - requires 2 sides (b & c) and the angle in-between (A)
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If calculating an angle (A) – requires all 3 sides (a, b & c)
To calculate a length use:
A
𝑎𝑎2 = 𝑏𝑏 2 + 𝑐𝑐 2 − 2𝑏𝑏𝑏𝑏 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
(Note – b & c is either side of the angle A)
c
b
To calculate an angle use:
𝑏𝑏 2 + 𝑐𝑐 2 − 𝑎𝑎2
𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 =
2𝑏𝑏𝑏𝑏
B
c
a
(Note – b & c is either side of the angle A)
Example One – Finding a length
9 cm
𝑥𝑥
28°
12 cm
2
2
2
𝑥𝑥 = 9 + 12 − 2 × 9 × 12 × cos 28
𝑥𝑥 2 = 34.28332
𝑥𝑥 = √34.28332
𝑥𝑥 = 5.9 𝑐𝑐𝑐𝑐 (𝑡𝑡𝑡𝑡 1 𝑑𝑑. 𝑝𝑝. )
 www.national5maths.co.uk 2015
Example Two – Finding an angle
21 cm
28 cm
33 cm
𝑐𝑐𝑜𝑜𝑜𝑜 𝑦𝑦 =
𝑦𝑦
282 + 332 − 212
2 × 28 × 33
𝑦𝑦 = 𝑐𝑐𝑐𝑐𝑐𝑐 −1 (0.774891)
𝑦𝑦 = 39.2° (1 𝑑𝑑. 𝑝𝑝. )