Question 1183392
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*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\tan(x)\,+\,\sqrt{3}\sin(x)\ =\ 0]


Factor the LHS:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\(\tan(x)\,+\,\sqrt{3}\)\ =\ 0]


Use the Zero Product Property:  If *[tex \Large ab\ =\ 0] then either *[tex \Large a\ =\ 0] or *[tex \Large b\ =\ 0].  Hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \sin(x)\ =\ 0]


or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \tan(x)\ +\ sqrt{3}\ =\ 0]


Use the unit circle to find the angles where *[tex \Large \sin(x)\ =\ 0] and *[tex \Large \tan(x)\ =\ -\sqrt{3}] in the interval *[tex \Large 0\ \leq\ x\ <\ 2\pi].  Then add the appropriate periodicity so that all possible angles are represented.

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

From <https://www.algebra.com/cgi-bin/upload-illustration.mpl> 
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