Question

If $4\sin x.\cos y+2\sin x+2\cos y+1=0$ where $x,y\in \left[0,2\pi \right]$ find the largest value of the sum (x+y).

Answer 1

$4\sin x\cos y+2\sin x+2\cos y+1=0$

$(2\sin x+1)(2\cos y+1)=0$

$\sin x=-\frac{1}{2}$ or $\cos y=\frac{-1}{2}$

$x=\frac{7\pi }{6},\frac{11\pi }{6}$ or $y=\frac{2\pi }{3},\frac{4\pi }{3}$

Largest value of $x+y=\frac{11\pi }{6}+\frac{4\pi }{3}=\frac{19\pi }{6}$