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The cosine rule to find missing sides | Pearson, Study notes of Analytical Geometry

1. Work out the length of the unknown side in each triangle. Give your answers correct to 3 significant figures. a b c d. Answers.

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A2400 ch9a | Version 1.1 | September 2020
The cosine rule to find missing sides
A LEVEL LINKS
Scheme of work: 4a. Trigonometric ratios and graphs
Key points
a is the side opposite angle A.
b is the side opposite angle B.
c is the side opposite angle C.
You can use the cosine rule to find the length of a side when two sides and the included
angle are given.
To calculate an unknown side use the formula .
Alternatively, you can use the cosine rule to find an unknown angle if the lengths of all three
sides are given.
To calculate an unknown angle use the formula .
Example 1 Work out the length of side w.
Give your answer correct to 3 significant figures.
w2 = 33.804 040 51...
w =
w = 5.81 cm
1 Always start by labelling the angles
and sides.
2 Write the cosine rule to find the
side.
3 Substitute the values a, b and A into
the formula.
4 Use a calculator to find w2 and
then w.
5 Round your final answer to 3
significant figures and write the
units in your answer.
222
2cosabc bc A=+-
222
cos
2
bca
A
bc
+-
=
222
2cosabc bc A=+-
222
87287cos45w=+-´´´ °
33.804 04051
pf2

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A 24 00 ch 9 a | Version 1.1 | September 2020

The cosine rule to find missing sides

A LEVEL LINKS

Scheme of work: 4a. Trigonometric ratios and graphs

Key points

  • a is the side opposite angle A. b is the side opposite angle B. c is the side opposite angle C.
  • You can use the cosine rule to find the length of a side when two sides and the included angle are given.
  • To calculate an unknown side use the formula.
  • Alternatively, you can use the cosine rule to find an unknown angle if the lengths of all three sides are given.
  • To calculate an unknown angle use the formula. Example 1 Work out the length of side w. Give your answer correct to 3 significant figures. w^2 = 33.804 040 51... w = w = 5.81 cm 1 Always start by labelling the angles and sides. 2 Write the cosine rule to find the side. 3 Substitute the values a , b and A into the formula. 4 Use a calculator to find w^2 and then w. 5 Round your final answer to 3 significant figures and write the units in your answer. a^2 = b^2^ + c^2 - 2 bc cos A 2 2 2 cos 2 b c a A bc

a^2^ = b^2 + c^2^ - 2 bc cos A w^2^ = 82 + 72 - 2 ´ 8 ´ 7 ´ cos 45°

A 24 00 ch 9 a | Version 1.1 | September 2020 Example 2 Work out the size of angle θ. Give your answer correct to 1 decimal place. θ = 122.878 349... θ = 122.9° 1 Always start by labelling the angles and sides. 2 Write the cosine rule to find the angle. 3 Substitute the values a , b and c into the formula. 4 Use cos−1^ to find the angle. 5 Use your calculator to work out cos–^1 (–76 ÷ 140). 6 Round your answer to 1 decimal place and write the units in your answer. Practice questions 1 Work out the length of the unknown side in each triangle. Give your answers correct to 3 significant figures. a b c d Answers 6 a 6.46 cm b 9.26 cm c 70.8 mm d 9.70 cm 2 2 2 cos 2 b c a A bc

cos 2 10 7 q

76 cos 140 q

=