Ratio of Sines

Geometry Level 3

In triangle A B C ABC , we have sin ⁡ A : sin ⁡ B : sin ⁡ C = 5 : 7 : 9 \sin A : \sin B : \sin C = 5:7:9 .

Find cos ⁡ ( A + B ) \cos (A+B) .


The answer is 0.1.

Alexander Koran by Alexander Koran , 21, USA

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1 solution

Alexander Koran
Jul 16, 2019

In order to follow the ratio of sines, we let A B = 9 , B C = 5 , A C = 7 AB = 9, BC = 5, AC = 7 . Law of Sines shows the ratio to be true.

Thus, cos ⁡ ( A + B ) = cos ⁡ ( 180 − C ) = − cos ⁡ ( C ) \cos(A+B) = \cos(180-C) = -\cos(C) .

Law of Cosines gives us − cos ⁡ ( C ) = − 5 2 + 7 2 − 9 2 2 ⋅ 5 ⋅ 7 = 1 10 -\cos(C) = -\frac{5^2+7^2-9^2}{2 \cdot 5 \cdot 7} = \boxed{\frac{1}{10}}

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