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Trigonometry : Key Stage 5 (AS-Level) : Mathematics Content Knowledge


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Key Stage 5 (AS-Level)
Trigonometry
Question 1 of 11

1. How confident are you that you can use the cosine rule to:

a. find missing lengths and angles of a non-right-angled triangle?


Example

Cosine rule: in any triangle ABC with lengths a, b, c:

c2 = a2 + b2 – 2ab cos C

The labelling of the triangle is arbitrary except that a is the length opposite vertex Ab is the length opposite vertex B and so on.

Generic triangle

The cosine rule can also be written as:

     a2 = b2 + c2 – 2bc cos A
or b2 = a2 + c2 – 2ac cos B

Example 1:
In triangle ABC, AC = 17cm, BC = 9cm and angle ACB = 20°. Calculate length AB.

Using the cosine rule gives

AB2 = 92 + 172 – 2 × 9 × 17 cos 20°
AB2 = 82.45
AB = 9.08 cm which is the length to 2dp.

Note that the cosine rule works for any triangle including right-angled triangles. When you have a right-angled triangle the cosine rule defaults to Pythagoras’ Theorem.

Right-angled triangle

Example 2:
In triangle ABC:
a = 12 cm
b = 6 cm
c = 8 cm

Find cos B, giving your answer in exact form.

From the cosine rule:
b2 = a2 + c2 – 2ac cos B

This gives 62 = 122 + 82 – 2 × 12 × 8 cos B

Therefore
36 = 144 + 64 – 182 cos B
36 = 208 – 192 cos B
–172 = –192 cos B
cos B =

172
192
=
43
48

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