Question

If $n$ be the number of solution of the equation $3co{s}^{2}x-8sinx=0$ in $\left[0,3\pi \right]$ is, then the value of $n$ is equal to

Answer 1

Answer: 4

We have,
$\begin{array}{l}3{\cos }^{2}x-8\sin x=0\\ \Rightarrow 3\left(1-{\sin }^{2}x\right)-8\sin x=0\\ \Rightarrow 3-3{\sin }^{2}x-8\sin x=0\\ \Rightarrow 3{\sin }^{2}x+8\sin x-3=0\\ \Rightarrow 3{\sin }^{2}x+9\sin x-\sin x-3=0\\ \Rightarrow 3\sin x(\sin x+3)-(\sin x+3)=0\\ \Rightarrow (3\sin x-1)(\sin x+3)=0\\ \Rightarrow \sin x=\frac{1}{3},-3\end{array}$
But $\sin x\in [-1,1]$ for all $x\in R$. Hence,
$\sin x=\frac{1}{3}$