anonymous
  • anonymous
Which of the following are solutions to the equation cos^2 (2x) - 1/4 = 0? Check ALL that apply. A. 5 pi/ 6 B. 11pi / 3 C. pi / 6 D. 12pi / 6
Mathematics
chestercat
  • chestercat
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anonymous
  • anonymous
From doing some current calculations. I came to the conclusion of C. Not sure if it is right or if there are anymore answers
SolomonZelman
  • SolomonZelman
\(\large\color{black}{ \displaystyle {\rm Cos}^2 (2x) - \frac{1}{4} = 0 }\) \(\large\color{black}{ \displaystyle {\rm Cos}^2 (2x)= \frac{1}{4} }\) \(\large\color{black}{ \displaystyle \left(~~ {\rm Cos} (2x)~\right)^2= \left(\frac{1}{2}\right)^2 }\)
SolomonZelman
  • SolomonZelman
hope that helps...

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SolomonZelman
  • SolomonZelman
(take the square root of both sides, and don't forget the \(\pm\) )
anonymous
  • anonymous
Ok! I will try to solve from there and I'll tell you what I get
SolomonZelman
  • SolomonZelman
sure:)
SolomonZelman
  • SolomonZelman
(oh, there is only 1 incorrect option)
anonymous
  • anonymous
So \[\cos(2x) = \pm \frac{ 1 }{ 2 }\] If I remember how to do this properly, I think I find what 1/2 is on the unit circle, which would be 60? Then divide 60 by 2 and get 30 degrees?
anonymous
  • anonymous
So there are 3 correct answers?
SolomonZelman
  • SolomonZelman
yes
SolomonZelman
  • SolomonZelman
30 degrees is one of them (as you said)
anonymous
  • anonymous
Ok! Now I do not know where to do from here
anonymous
  • anonymous
We already have \[\frac{ \pi }{ 6}\] as one answer
SolomonZelman
  • SolomonZelman
you can plug the rest of the options, if you don't want to go ahead and generate the solution sets for -1/2 and +1/2.
SolomonZelman
  • SolomonZelman
yes π/6 is correct
anonymous
  • anonymous
I don't think I plugged the numbers in properly, but I got 12pi/6 and 5pi/6?
SolomonZelman
  • SolomonZelman
12π/6 is not right
SolomonZelman
  • SolomonZelman
5π/6 is right
anonymous
  • anonymous
Ok! So how would I plug them in? I know I did it wrong haha
SolomonZelman
  • SolomonZelman
(click alt, and hold it click 2 2 7 respectively on your number pad on the left if you have one release alt: you get π)
SolomonZelman
  • SolomonZelman
you would just plug the answer choices instead of x, into that equation cos²(2x)-1/4=0
anonymous
  • anonymous
Oh and what gets you = 0 right?
anonymous
  • anonymous
\[2 \cos ^22x=\frac{ 1 }{ 2 }\] \[1+\cos 4x=\frac{ 1 }{ 2 }\] \[\cos 4x=\frac{ 1 }{ 2 }-1=-\frac{ 1 }{ 2 }=-\cos \frac{ \pi }{ 3 }=\cos \left(2n \pi+ \pi \pm \frac{ \pi }{ 3 } \right)\] where n is an integer. \[4x=\left( 2n+1 \right)\pi \pm \frac{ \pi }{ 3 }\] \[x=\frac{ 1 }{ 4 }\left\{ \left( 2n+1 \right)\pi \pm \frac{ \pi }{ 3 } \right\}\] now you can check by plugging n=0,1,2,....
SolomonZelman
  • SolomonZelman
cos(4x) :) I haven't thought of that one....
anonymous
  • anonymous
So the answers would be \[\frac{ 5\pi }{ 6}, \frac{ 11\pi }{ 3 }, and \frac{ \pi }{ 6 }?\]
SolomonZelman
  • SolomonZelman
yes. use ~ for space
SolomonZelman
  • SolomonZelman
yw
anonymous
  • anonymous
Thank you for your time and the tips :)

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