Question

If $0\le x\le 3\pi ,0\le y\le 3\pi$ and $\cos x\sin y=1$, then the possible number of values of the ordered pair $(x,y)$ is

• A. $4$
• B. $5$
• C. $6$
• D. $7$

Answer 1

The correct option is C $6$

$\cos x\sin y=1\Rightarrow \cos x=1,\sin y=1\text{or}\cos x=-1,\sin y=-1$

Now,
$\cos x=1,\sin y=1\Rightarrow x=0,2\pi \text{and}y=\frac{\pi }{2},\frac{5\pi }{2}....\left(i\right)$

$\cos x=-1,\sin y=-1\Rightarrow x=\pi ,3\pi \text{and}y=\frac{3\pi }{2}....\left(ii\right)$

From equations $\left(i\right)$ and $\left(ii\right)$, the required ordered pairs are
$(0,\frac{\pi }{2}),(0,\frac{5\pi }{2}),(2\pi ,\frac{\pi }{2}),(2\pi ,\frac{5\pi }{2}),(\pi ,\frac{3\pi }{2}),(3\pi ,\frac{3\pi }{2})$
So, there are $6$ ordered pairs of $(x,y)$