Given-that-in-ABC-sin-A-sin-B-sin-B-sin-C-sin-C-sin-A-6-4-5-Find-the-angle-A-

Question Number 161464 by ZiYangLee last updated on 18/Dec/21

Given that in Δ ABC,

(\sin A + \sin B) : (\sin B + \sin C) : (\sin C + \sin A) = 6 : 4 : 5

Find the angle A .

Commented by cortano last updated on 18/Dec/21

{\begin{cases} \sin A + \sin B = 6 k \\ \sin B + \sin C = 4 k \\ \sin C + \sin A = 5 k \end{cases}

\Rightarrow \sin A + \sin B + \sin C = \frac{15 k}{2}

\Rightarrow {\begin{cases} \sin A = \frac{7 k}{2} \\ \sin B = \frac{5 k}{2} \\ \sin C = \frac{3 k}{2} \end{cases}; \sin A : \sin B : \sin C = 7 : 5 : 3

s o a : b : c = 7 : 5 : 3

\Rightarrow \cos A = \frac{25 + 9 - 49}{2 \times 5 \times 3} = - \frac{15}{30}

\Rightarrow A = 120 °

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