Answer:
[tex]x=\p \pi n ,x=\frac{3\pi}{2} \pm 2\pi n[/tex]
Step-by-step explanation:
We want to solve the equation,
[tex]sinx(sin x+1)=0[/tex]
By the zero product property of multiplication,
[tex]sinx=0\:or\:(sin x+1)=0[/tex]
[tex]\Rightarrow sinx=0\:or\:sin x=-1[/tex]
If [tex]sinx=0[/tex], then, [tex]x=\pi[/tex]
The general solution is
[tex]x=\pm \pi n[/tex]
For [tex]sinx=-1[/tex], it means [tex]x[/tex] is either in the third quadrant or fourth quadrant.
So we first solve for,
[tex]sinx=1[/tex]
This implies that,
[tex]x=\frac{\pi}{2}[/tex]
In the third quadrant,
[tex]x=\pi +\frac{\pi}{2}[/tex]
[tex]x=\frac{3\pi}{2}[/tex]
In the fourth quadrant,
[tex]x=2\pi -\frac{\pi}{2}[/tex]
[tex]x=\frac{3\pi}{2}[/tex]
This is a repeated solution.
So the general solution is
If [tex]sin x=-1[/tex], then [tex]x=\frac{3\pi}{2}\pm 2\pi n[/tex]
Putting the two solutions together gives
[tex]x=\pm \pi n ,x=\frac{3\pi}{2} \pm 2\pi n[/tex], where n is an integer
The correct answer is C.
