If $x$ and $y$ are acute angles such that $\cos x+\cos y=\frac{3}{2}$ and $\sin x+\sin y=\frac{3}{4}$, then $\sin (x+y)$ equals to

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Aicharya · Accepted Answer

Given, $\cos x+\cos y=\frac{3}{2}$ $ \Rightarrow \quad 2 \cos \left(\frac{x+y}{2}ight) \cos \left(\frac{x-y}{2}ight)=\frac{3}{2} $ and $ \begin{aligned} & \sin x+\sin y=\frac{3}{4} \ & \Rightarrow \quad 2 \sin \left(\frac{x+y}{2}ight) \cos \left(\frac{x-y}{2}ight)=\frac{3}{4} \ & 	herefore \quad \frac{2 \sin \left(\frac{x+y}{2}ight) \cos \left(\frac{x-y}{2}ight)}{2 \cos \left(\frac{x+y}{2}ight) \cos \left(\frac{x-y}{2}ight)}=\frac{3 / 4}{3 / 2} \ & \Rightarrow 	an \left(\frac{x+y}{2}ight)=\frac{1}{2} \ & 	herefore \sin (x+y)=\frac{2 	an \left(\frac{x+y}{2}ight)}{1+	an ^2\left(\frac{x+y}{2}ight)} \ & =\frac{2 	imes \frac{1}{2}}{1+\left(\frac{1}{2}ight)^2}=\frac{4}{4+1}=\frac{4}{5} \ & \end{aligned} $

If $x$ and $y$ are acute angles such that $\cos x+\cos y=\frac{3}{2}$ and $\sin x+\sin y=\frac{3}{4}$, then…

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