# How do you find all solutions of the equation sin(x+pi/6)-sin(x-pi/6)=1/2 in the interval [0,2pi)?

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#### Explanation:

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Oct 26, 2017

$x = \frac{\pi}{6}$ or $\frac{11 \pi}{6}$

#### Explanation:

We have the identity $\sin \left(A + B\right) - \sin \left(A - B\right) = 2 \cos A \sin B$

hence $\sin \left(x + \frac{\pi}{6}\right) - \sin \left(x - \frac{\pi}{6}\right) = \frac{1}{2}$ can be written as

$2 \cos x \sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$

or $2 \cos x \times \frac{1}{2} = \frac{1}{2}$

or $\cos x = \frac{1}{2} = \cos \left(\frac{\pi}{6}\right)$

and in the interval $\left[0 , 2 \pi\right)$

$x = \frac{\pi}{6}$ or $2 \pi - \frac{\pi}{6} = \frac{11 \pi}{6}$

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