Question

Consider the system of equations
$x{\cos }^{3}y+3x\cos y{\sin }^{2}y=14$
$x{\sin }^{3}y+3x{\cos }^{2}y\sin y=13$

The number of values of $y\in [0,6\pi ]$ is

• A. $5$
• B. $3$
• C. $4$
• D. $6$

Answer 1

The correct option is D $6$

$x{\cos }^{3}y+3x\cos y{\sin }^{2}y=14\cdots \left(1\right)$
$x{\sin }^{3}y+3x{\cos }^{2}y\sin y=13\cdots \left(2\right)$
Adding Eqs. $\left(1\right)$ and $\left(2\right)$
$x({\cos }^{3}y+3{\cos }^{2}y\sin y+3\cos y{\sin }^{2}y+{\sin }^{3}y)=27$
$\Rightarrow x(\cos y+\sin y{)}^3=27$
$\Rightarrow x^{1/3}(\cos y+\sin y)=3\cdots \left(3\right)$

Subtracting Eq $\left(2\right)$ from Eq $\left(1\right)$, we get
$x({\cos }^{3}y+3\cos y{\sin }^{2}y-3{\cos }^{2}y\sin y-{\sin }^{3}y)=1$
$\Rightarrow x(\cos y-\sin y{)}^3=1$
$\Rightarrow x^{1/3}(\cos y-\sin y)=1\cdots \left(4\right)$

Dividing Eq $\left(3\right)$ by Eq $\left(4\right)$, we get
$\cos y+\sin y=3\cos y-3\sin y$
$\Rightarrow \tan y=\frac{1}{2}$

$\tan$ function is positive in $1$st quadrant and $3$rd quadrant.
Therefore, there are exactly six values of $y\in [0,6\pi ]$