To solve the equation \(\sin x - 2 \sin x \cos x = 0\) on the interval \([0, 2\pi)\), let's follow a step-by-step approach:
1. Factor the Equation:
[tex]\[
\sin x - 2 \sin x \cos x = 0
\][/tex]
We can factor out \(\sin x\) from the left-hand side:
[tex]\[
\sin x (1 - 2 \cos x) = 0
\][/tex]
2. Set Each Factor to Zero:
We have two factors: \(\sin x\) and \((1 - 2 \cos x)\). We set each factor to zero separately and solve for \(x\).
First Factor: \(\sin x = 0\)
[tex]\[
\sin x = 0
\][/tex]
The solutions for \(\sin x = 0\) in the interval \([0, 2\pi)\) are:
[tex]\[
x = 0, \pi
\][/tex]
Second Factor: \(1 - 2 \cos x = 0\)
[tex]\[
1 - 2 \cos x = 0
\][/tex]
Solve for \(\cos x\):
[tex]\[
2 \cos x = 1 \implies \cos x = \frac{1}{2}
\][/tex]
The solutions for \(\cos x = \frac{1}{2}\) in the interval \([0, 2\pi)\) are:
[tex]\[
x = \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
3. Combine All Solutions:
Combining the solutions from both factors, we get:
[tex]\[
x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}
\][/tex]
Therefore, the correct choice is:
A. \(x = 0, \pi, \frac{\pi}{3}, \frac{5\pi}{3}\)
These are the exact answers for the solutions to the equation [tex]\(\sin x - 2 \sin x \cos x = 0\)[/tex] on the interval [tex]\([0, 2\pi)\)[/tex].
