Solving Triangles using the Law of Cosines

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Solving Triangles using the Law of Cosines

• Mrs. Keller

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District Objective 7-D

• Students will apply the Law of Cosines to solve
  triangles
• MA GSR 4 B 11
• MA GSR 4 A 12
• Concept map

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We have already discussed situations where the
Law of Sines applies to the given information.

• Such situations include
• ASA
• SAA
• SSA (the ambiguous case)

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• We will now discuss situations where the
• given information implies
• SAS
• SSS

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Suppose we are given
B
c 2
a
45
C
A
b 4
WE CANNOT USE LAW OF SINES!!!!! Sin A Sin B
Sin C a b
c The information does not fit!
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We Need the Law of Cosines!!!

• The Law of Cosines comes in 3 similar forms.
• 1) c² a² b² - 2ab cos C
• 2) b² a² c² -2ac cos B
• 3) a² b² c² -2bc cos A

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We can use a² b² c² -2bc cos A and find the
side length a
B
c 2
a
45
C
A
b 4
Substituting we obtain a² 4² 2² -2(4)(2) cos
(45) a²
16 4 16 ( .7071)
a² 20 11.3136
a² 8.6864
a approx. 2.95
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Now, we can use the Law of Sines to find angle C
B
Sin 45 Sin C 2.95
2 .7071 Sin C 2.95
2 Sin C approx. .4794 C approx 28.6
c 2
a 2.95
45
C
A
b 4
Since the sum of the angles totals 180, then B
106.4