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
Algebra.Com
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 \ +\ \sin^2(x)\ =\ 1)
Which means that you can re-cast your equation thus:
)\ =\ -3\sin(x))
and then a little re-arranging:
\ -\ 3\sin(x)\ -\ 2\ =\ 0)
Next, let )
, so:
which factors rather tidily:
(2u\ +\ 1)\ =\ 0)
Hence
or
But recall that . Since the range of )
is \ \leq\ 1\right])
, we must exclude 
. Therefore:
\ =\ -\frac{1}{2})
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
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