Question

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 value of $\sin (x+y)$.

Answer 1

Given $cosx+cosy=\frac{3}{2}$ __ (i)

$sinx+siny=\frac{3}{4}$ __ (ii)

$sinx+siny=\frac{3}{4}$

$\Rightarrow 2sin\left(\frac{x+y}{2}\right)cos\left(\frac{x-y}{2}\right)=\frac{3}{4}$ __ (iii)

$cosx+cosy=\frac{3}{2}$

$2cos\left(\frac{x+y}{2}\right)cos\left(\frac{x-y}{2}\right)=\frac{3}{2}$ __ (iv)

Dividing (iii) by (iv)

$tan\left(\frac{x+y}{2}\right)=\frac{1}{2}$ __ (v)

$\therefore sin\left(\frac{x+y}{2}\right)=\frac{1}{\sqrt{5}}$

$cos\left(\frac{x+y}{2}\right)=\frac{2}{\sqrt{5}}$

$sin(x+y)=2sin\left(\frac{x+y}{2}\right)cos\left(\frac{x+y}{2}\right)$

$=2\times \frac{1}{\sqrt{5}}\times \frac{2}{\sqrt{5}}$

$sin(x+y)=\frac{4}{5}$