Question 640256
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin\left(2x\right)\ =\ \sin\left(x\right)]


Use the double angle formula:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\sin(x)cos(x)\ =\ \sin(x)]


Multiply both sides by *[tex \LARGE \frac{1}{\sin(x)}]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2\cos(x)\ =\ 1]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \cos(x)\ =\ \frac{1}{2}]


Use the unit circle to find the two values of *[tex \LARGE x] that result in *[tex \LARGE \cos(x)\ =\ \frac{1}{2}] in the specified interval (once around the circle).  Remember, *[tex \LARGE \cos(x)] is the *[tex \LARGE x] coordinate of the point of intersection of the terminal ray and the unit circle.


<img src="http://www.math.ucsd.edu/~jarmel/math4c/Unit_Circle_Angles.png">


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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