Let x ε[–π, π]. S = {x : sin x(sin x + cos x) = a, aεI} Then number of elements in set S is equal to

The correct option is 9.

To find the number of elements in the set $S=\{x:sinx(sinx+cosx)=a,a\in I\}$ for $x\in [-\pi ,\pi ]$, we first analyze the given equation:

$f\left(x\right)=sinx(sinx+cosx)={sin}^{2}x+sinxcosx$

Using standard trigonometric identities, we can rewrite the terms as follows:
${sin}^{2}x=\frac{1-cos2x}{2}$
$sinxcosx=\frac{sin2x}{2}$

Substituting these identities back into the expression for $f\left(x\right)$ gives:
$f\left(x\right)=\frac{1-cos2x}{2}+\frac{sin2x}{2}$
$f\left(x\right)=\frac{1}{2}+\frac{1}{2}(sin2x-cos2x)$

Now, we find the range of the function $f\left(x\right)$ over the interval $x\in [-\pi ,\pi ]$. We know that:
$sin2x-cos2x=\sqrt{2}sin(2x-\frac{\pi }{4})$

Since the interval for $x$ covers more than a full period of the sine function, the term $sin(2x-\frac{\pi }{4})$ attains all values in the interval $[-1,1]$. Therefore, the range of $f\left(x\right)$ is:
$[\frac{1}{2}-\frac{\sqrt{2}}{2},\frac{1}{2}+\frac{\sqrt{2}}{2}]$

Using the approximation $\sqrt{2}\approx 1.414$, the range of $f\left(x\right)$ is approximately:
$[0.5-0.707,0.5+0.707]=[-0.207,1.207]$

Since $a$ must be an integer ($a\in I$), the only possible integer values for $a$ in this range are:
$a=0$ and $a=1$

Let us find the solutions for each case within the interval $x\in [-\pi ,\pi ]$:
Case 1: $a=0$
$sinx(sinx+cosx)=0$
This yields two possibilities:
1) $sinx=0\Rightarrow x=-\pi ,0,\pi$ (3 solutions)
2) $sinx+cosx=0\Rightarrow tanx=-1\Rightarrow x=-\frac{\pi }{4},\frac{3\pi }{4}$ (2 solutions)
Thus, Case 1 gives a total of $3+2=5$ distinct solutions.

Case 2: $a=1$
${sin}^{2}x+sinxcosx=1$
Using the identity $1-{sin}^{2}x={cos}^{2}x$, we have:
$sinxcosx={cos}^{2}x$
$cosx(sinx-cosx)=0$
This yields two possibilities:
1) $cosx=0\Rightarrow x=-\frac{\pi }{2},\frac{\pi }{2}$ (2 solutions)
2) $sinx-cosx=0\Rightarrow tanx=1\Rightarrow x=-\frac{3\pi }{4},\frac{\pi }{4}$ (2 solutions)
Thus, Case 2 gives a total of $2+2=4$ distinct solutions.

Since all the solutions from Case 1 and Case 2 are mutually exclusive and lie within the interval $[-\pi ,\pi ]$, the total number of elements in set $S$ is:
$5+4=9$