In any - ABC -sinA-sin B-sin C-sin A- 3 then show that the equality holds if and only if triangle is equilateral-

We know that #160; asinA=bsinB=csinC ...(sine rule).Substituting in the above expression, we get #160; √(a)√(b)+√(c) √(a)+√(b)√(a)+√(c) √(b)+√(c)√( ...

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Byju's Answer

Standard XII

Mathematics

Properties of Argument

In any Δ AB...

Question

In any $Δ A B C$ $\sum \frac{\sqrt{sin} A}{\sqrt{sin B} + \sqrt{sin C} - \sqrt{sin A}} \geq 3$ then show that the equality holds if and only if triangle is equilateral.

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Solution

We know that
$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$ ...(sine rule).

Substituting in the above expression, we get

$\frac{\sqrt{a}}{\sqrt{b} + \sqrt{c} - \sqrt{a}} + \frac{\sqrt{b}}{\sqrt{a} + \sqrt{c} - \sqrt{b}} + \frac{\sqrt{c}}{\sqrt{a} + \sqrt{b} - \sqrt{c}}$

Now if it is an equilateral triangle,
$a = b = c$

Hence the above expression reduces to

$\frac{\sqrt{a}}{\sqrt{b} + \sqrt{c} - \sqrt{a}} + \frac{\sqrt{b}}{\sqrt{a} + \sqrt{c} - \sqrt{b}} + \frac{\sqrt{c}}{\sqrt{a} + \sqrt{b} - \sqrt{c}}$
$= 3 [\frac{\sqrt{a}}{\sqrt{a} + \sqrt{a} - \sqrt{a}}]$

$= 3$

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In any ΔABC ∑√sinA√sinB+√sinC−√sinA≥3 then show that the equality holds if and only if triangle is equilateral.

In any $Δ A B C$ $\sum \frac{\sqrt{sin} A}{\sqrt{sin B} + \sqrt{sin C} - \sqrt{sin A}} \geq 3$ then show that the equality holds if and only if triangle is equilateral.