Question

The maximum integral value of $a$ for which the equation $a\sin x+\cos 2x=2a-7$ has a solution

Answer 1

Consider the problem

$cos2x$ can be written as $1-2{\sin }^{2}x$ so the equation becomes

$\begin{array}{c}1-2{\sin }^{2}x+a\sin x+7-2a=0\\ 2{\sin }^{2}x-a\sin x-8+2a=0\end{array}$

It is quadratic in $sinx$,

So by the using quadratic Formula we get

$\sin x=a\pm \sqrt{\frac{a^2-4\times 2\left(2a-8\right)}{4}}$

On solving we get

$\sin x=\frac{\left(a-4\right)}{2}$

To find the possible values of $a$ we will use the following in equation

As we know that value of $sinx$ lies between $-1$ to $1$

$\begin{array}{c}-1\le \frac{\left(a-4\right)}{2}\le 1\\ -2\le \left(a-4\right)\le 2\\ 2\le a\le 6\end{array}$

So, for this range the solution of the trigonometric equation exists