Trigonometric Ratios Worksheet Page 19
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221.6 Solving Problems Using the Cosine Law
KEY CONCEPTS
• The cosine law states that for any △ABC,
A
2
2
2
a
= b
+ c
− 2bc cos A
c
b
2
2
2
= a
+ c
− 2ac cos B
b
B
C
a
2
= a
2
+ b
2
− 2ab cos C
c
• A side length can be determined if the lengths of the other two sides and the
measure of the contained angle are known.
• An angle measure can be determined if all three side lengths are known.
Example
C
The lengths of the sides of a triangular property on Sydenham
Lake are 62.0 ft, 87.8 ft, and 118.8 ft. Determine the angles
formed between the sides of the triangular property to the
118.8 ft
87.8 ft
nearest tenth of a degree.
Solution
A
B
For △ABC, the side lengths are a = 118.8, b = 87.8, and c = 62.0.
62.0 ft
Use the cosine law to determine ∠A.
2
2
2
b
+ c
− a
___________
cos A =
2bc
2
2
2
87.8
+ 62.0
− 118.8
____________________
=
2(87.8)(62.0)
= −0.2351� �
∠A = 103.6030�°
Check for the ambiguous case. Given ∠A and a > b, only one triangle is possible.
Use the sine law to determine ∠B.
sin B
sin 103.6030�°
_____
______________
=
87.8
118.8
87.8 sin 103.6030�°
__________________
sin B =
118.8
= 0.7183�
∠B = 45.9164� °
Determine ∠C.
∠C = 180° − (103.6030�° + 45.9164�°)
= 30.4805�°
≐ 30.5°
978-0-07-090893-2 Mathematics for College Technology 12 Study Guide and Exercise Book • MHR 19
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