Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$I f i n a Δ A B C, \frac{c o s A}{a} = \frac{c o s B}{b} = \frac{c o s C}{c}, then find the measures of angles A, B, C.$

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Solution

$\frac{c o s A}{a} = \frac{c o s B}{b} = \frac{c o s C}{c} \Rightarrow \frac{b^{2} + c^{2} - a^{2}}{2 b c} = \frac{a^{2} + c^{2} - b^{2}}{2 a b c} = \frac{a^{2} + b^{2} - c^{2}}{2 a b c} \Rightarrow b^{2} + c^{2} - a^{2} = a^{2} + c^{2} - b^{2} = a^{2} + b^{2} - c^{2} ∴ b^{2} + c^{2} - a^{2} = a^{2} + c^{2} - b^{2} 2 b^{2} = 2 a^{2} \Rightarrow b = a ∴ a^{2} + c^{2} - b^{2} = a^{2} + b^{2} - c^{2} 2 c^{2} = 2 b^{2} \Rightarrow c = b b^{2} + c^{2} - a^{2} = a^{2} + b^{2} - c^{2} \Rightarrow 2 c^{2} = 2 a^{2} \Rightarrow c = a H e n c e a = b = c o r A = B = C = 60^{\circ}$

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