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Definition of Functions

Let a,b and...

Question

Let $a, b$ and $c$ be the sides of a $△ A B C$ . If $a^{2}, b^{2}, c^{2}$ are the roots of the equation $x^{3} - P x^{2} + Q x - R = 0$ , where $P, Q, R$ are constants, then find the value of $\frac{cos A}{A} + \frac{cos B}{B} + \frac{cos C}{C}$ in terms of $P, Q$ and $R$ .

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Solution

Given, $\frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c}$ ---------------------------(1)
Applying cosine rule:
$cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}, cos B = \frac{a^{2} + c^{2} - b^{2}}{2 a c}, cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b}$
putting values in equation (1)
= $\frac{b^{2} + c^{2} - a^{2}}{2 a b c} + \frac{a^{2} + c^{2} - b^{2}}{2 a b c} + \frac{a^{2} + b^{2} - c^{2}}{2 a b c}$
= $\frac{(a^{2} + b^{2} + c^{2})}{2 a b c}$
Now,
$X^{3} - P x^{2} + Q x - R = 0$
Sum of roots, $a^{2} + b^{2} + c^{2} = P$
Product of roots, $a^{2} b^{2} c^{2} = R$
Then, $\frac{(a^{2} + b^{2} + c^{2})}{2 a b c}$ = $\frac{p}{2 \sqrt{R}}$
$∴ \frac{cos A}{a} + \frac{cos B}{b} + \frac{cos C}{c} = \frac{P}{2 \sqrt{R}}$

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