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Trigonometric Ratios for Sum of Two Angles

If R+ r= r3 t...

Question

If $R + r = r_{3}$ then $∠ C =$

90^{\circ}

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60^{\circ}

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45^{\circ}

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120^{\circ}

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Solution

The correct option is B $60^{\circ}$
Given, $R + r = r_{3}$
$\Rightarrow r_{3} - r = R$
$\Rightarrow 4 R sin \frac{C}{2} (cos \frac{A}{2} \cdot cos \frac{B}{2} - sin \frac{A}{2} \cdot sin \frac{B}{2}) = R$
$\Rightarrow 4 R sin \frac{C}{2} \cdot cos (\frac{A + B}{2}) = R$
$\Rightarrow 4 R sin \frac{C}{2} \cdot cos (90^{\circ} - \frac{C}{2}) = R$
$\Rightarrow 4 R {sin}^{2} \frac{C}{2} = R$
$\Rightarrow {sin}^{2} \frac{C}{2} = \frac{1}{4}$
$\Rightarrow sin \frac{C}{2} = \frac{1}{2}$
$\Rightarrow ∠ C = 60^{\circ}$

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△ A B C

a < b < c

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