# Solving a trigonometric function

How do you solve this trigonometric equation?

Solve the equation for solutions in the interval $[0,2 \pi)$. $$\left(\cot(x) -1 \right) \left( 2 \sin(x) + \sqrt{3} \right) = 0.$$

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Hint: if $a \cdot b = 0$, then $a = 0$ or $b = 0$. Solve both equations to get all solutions. – Ayman Hourieh May 6 '12 at 23:35
Any particular reason you need to use offensive language in your user name? – Arturo Magidin May 6 '12 at 23:38
The profanity in the user's display name has been removed. @I'm just some guy: Any future use of profanity will be cause for suspension. – Zev Chonoles May 6 '12 at 23:50

## 1 Answer

A product of two real valued quantities is zero if and only if one of the terms in the product is zero.

So, here, either $$\cot x-1=0$$ or $$2\sin x +\sqrt 3=0.$$ Equivalently either $\cot x=1$ or $\sin x =-\sqrt 3/2$. Can you find the solutions to these? Remember to only take solutions in $[0,2\pi)$.

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Yep, I found the correct solution set as {pi/4, 5pi/4, 4pi/3, 5pi/3}. Thanks! – Austin May 6 '12 at 23:51