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Sine Rule

a cos A + b c...

Question

$a c o s A + b c o s B + c c o s C = 2 b s i n A s i n C = 2 c s i n A s i n B$

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Solution

$a c o s A + b c o s B + c c o s C = 2 b s i n A s i n C = 2 c s i n A s i n B L H S a c o s a + b c o s B + c c o s C = k s i n A c o s A + k s i n B c o s B + k s i n C c o s C = \frac{k}{2} (s i n 2 A + s i n 2 B + s i n 2 C) = \frac{k}{2} (2 s i n (A + B) . c o s (A - B) + 2 s i n C . c o s C) = \frac{2 k}{2} (s i n (π - C) . c o s (A - B) + s i n C . c o s C) = k (s i n C . c o s (A - B) + s i n C . c o s C) = k s i n C (c o s (A - B) + c o s C) = k s i n C .2 c o s (\frac{A - B + C}{2}) . c o s (\frac{A - B - C}{2}) = k s i n C .2 c o s (\frac{π - 2 B}{2}) . c o s (\frac{A - π + A}{2}) = k s i n C .2 s i n B . c o s (\frac{2 A - π}{2}) = k s i n C .2 s i n B . c o s (\frac{π - 2 A}{2}) = k s i n C .2 s i n B . s i n A = 2 s i n B s i n C (k s i n A) = 2 a s i n B s i n C = R H S Similarly, a c o s A + b c o s B + a c o s C = 2 c s i n A s i n B$

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