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General Solution of Trigonometric Equation

If R,r deno...

Question

If $R, r$ denote respectively the radii of the circumcircle and incircle of a triangle $A B C$ , then

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Solution

(a) $\frac{a}{s i n A} = 2 R$
$∴$ $\sum a c o s A = \sum 2 R s i n A c o s A = R \sum s i n 2 A = R$ $(4 s i n A s i n B s i n C)$
(b) $a = 2 R s i n A$
$\sum a c o t A = 2 R \sum c o s A = 2 R (1 + \frac{r}{R})$
$= 2 (R + r)$
(c) $4 R s i n \frac{A}{2} s i n \frac{B}{2} s i n \frac{C}{2} = r$
(d) $\sum \frac{a}{s i n A} = 2 R + 2 R + 2 R = 6 R$

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