SOLUTION: Find all solutions for the equation. Record your answers in exact form where possible. 2 cos ^2 x = -3sin x Thanks for your help, any help is appreciated

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Question 324020: Find all solutions for the equation. Record your answers in exact form where possible.

2 cos ^2 x = -3sin x
Thanks for your help, any help is appreciated
Answer by solver91311(24713) (Show Source):
You can put this solution on YOUR website!

First thing to do is remember that

Which means that you can re-cast your equation thus:

and then a little re-arranging:

Next, let , so:

which factors rather tidily:

Hence

or

But recall that . Since the range of is , we must exclude . Therefore:

Finally,

Looking at the unit circle and remembering that sin is the -coordinate of a point on the unit circle,

(see http://en.wikipedia.org/wiki/Unit_circle)

You see that there are two possibilities, or . However, the instructions for the problem say to find all solutions and does not give a range, hence we need to allow for the periodicity of the sine function. Therefore the final answer is:

or

where

John