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Solve the equation sin 2x - sin x = 0

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Answer to a math question Solve the equation sin 2x - sin x = 0

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Fred
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118 Answers
To solve the equation sin 2x - sin x = 0, we will use the trigonometric identity sin 2x = 2sin x cos x.

Substitute sin 2x = 2sin x cos x into the equation:
2sin x cos x - sin x = 0

Factor out sin x:
sin x(2cos x - 1) = 0

Now, we have two possibilities:
1) sin x = 0
2) 2cos x - 1 = 0

For sin x = 0:
x = kฯ€, where k is an integer.

For 2cos x - 1 = 0:
2cos x - 1 = 0
2cos x = 1
cos x = 1/2

Using the unit circle or the knowledge of cosine function values, we find x = ฯ€/3 + 2ฯ€k or x = 5ฯ€/3 + 2ฯ€k, where k is an integer.

Therefore, the solutions are:
x = kฯ€, x = ฯ€/3 + 2ฯ€k, x = 5ฯ€/3 + 2ฯ€k, where k is an integer.

\boxed{x = k\pi, \quad x = \frac{\pi}{3} + 2k\pi, \quad x = \frac{5\pi}{3} + 2k\pi, \quad \text{where } k \text{ is an integer}}

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