Angle $α$ is divided into two parts $A$ and $B$ such that $A - B = x$ and $tan A : tan B = p : q$ . The value of $sin x$ is equal to

\frac{(p + q) sin α}{p - q}

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\frac{p sin α}{p + q}

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\frac{p sin α}{p - q}

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\frac{(p - q) sin α}{p + q}

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Solution

The correct option is D $\frac{(p - q) sin α}{p + q}$
$α = A + B a n d x = A - B$

$A = \frac{x + α}{2}, B = \frac{α - x}{2} \frac{tan A}{tan B} = \frac{tan (\frac{x + α}{2})}{tan (\frac{α - x}{2})} = \frac{p}{q} \Rightarrow \frac{2 sin (\frac{α + x}{2}) c o s (\frac{α - x}{2})}{2 sin (\frac{α - x}{2}) c o s (\frac{α + x}{2})} = \frac{p}{q} \Rightarrow \frac{sin α + sin x}{sin α - sin x} = \frac{p}{q} \Rightarrow \frac{p + q}{p - q} = \frac{2 sin α}{2 sin x} \Rightarrow sin x = \frac{(p - q) sin α}{p + q}$

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Angle α is divided into two parts A and B such that A−B=x and tanA:tanB=p:q. The value of sinx is equal to

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Angle $α$ is divided into two parts $A$ and $B$ such that $A - B = x$ and $tan A : tan B = p : q$ . The value of $sin x$ is equal to