If-in-a-triangle-ABC-2-cos-A-a-cos-B-b-2-cos-C-c-a-bc-b-ca-then-the-value-of-the-angle-A-is-

Question Number 79453 by Vishal Sharma last updated on 25/Jan/20

If in a triangle A B C

2 \frac{\cos A}{a} + \frac{\cos B}{b} + 2 \frac{\cos C}{c} = \frac{a}{b c} + \frac{b}{c a}

then the value of the angle A is

Answered by $@ty@m123 last updated on 25/Jan/20

2 \frac{\cos A}{a} + \frac{\cos B}{b} + 2 \frac{\cos C}{c} = \frac{a}{b c} + \frac{b}{c a}

\Rightarrow 2 \frac{b^{2} + c^{2} - a^{2}}{2 a b c} + \frac{c^{2} + a^{2} - b^{2}}{2 a b c} + 2 \frac{a^{2} + b^{2} - c^{2}}{2 a b c} = \frac{a^{2} + b^{2}}{a b c}

\Rightarrow 2 (b^{2} + c^{2} - a^{2}) + c^{2} + a^{2} - b^{2} + 2 (a^{2} + b^{2} - c^{2}) = 2 (a^{2} + b^{2})

\Rightarrow a^{2} + 3 b^{2} + c^{2} = 2 a^{2} + 2 b^{2}

\Rightarrow a^{2} = b^{2} + c^{2}

\Rightarrow ∠ A = 90^{o}