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Solve the following equation on the interval \([0, 2\pi)\):

[tex]\[
\sin x - 2 \sin x \cos x = 0
\][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your response:

A. \(x =\) \(\square\)
(Type an exact answer, using \(\pi\) as needed. Use a comma to separate answers.)

B. There is no solution.



Answer :

GinnyAnswer
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].

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