anonymous
  • anonymous
How do I convert: y = 6 sqrt(2) sin(x)-6 sqrt(2) cos(x) into the form y = A sin(Bx-C) that has the same graph as the above? Wolfram|Alpha tells me I'm supposed to get: y = -12sin(pi/4 - x) But I want to know how I get there.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
im waiting for some one to respond to see how they figure this out!
anonymous
  • anonymous
Me too..
anonymous
  • anonymous
Original Equation for better readability: \[y = 6 \sqrt(2) \sin(x) - 6 \sqrt(2) \cos(x)\] And the equation in the \[y=A \sin(Bx-C)\] form: \[y = -12\sin(\pi/4 - x)\] ....How do I get from original to final equation posted?

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More answers

anonymous
  • anonymous
You want to put a sum of sine and cosine into one sine. You can consider the following:\[r \sin(x + \alpha) = r \cos \alpha \sin x + r \sin a \cos x\]
anonymous
  • anonymous
For the two expressions to be identical, you have to equate the coefficients on either side and solve for r and alpha.
anonymous
  • anonymous
Does this make sense?
anonymous
  • anonymous
So\[r \cos \alpha = 6 \sqrt{2}, r \sin \alpha = -6 \sqrt{2}\]
anonymous
  • anonymous
Divide sine by cosine to give\[\frac{r \sin \alpha}{r \sin \alpha}=\tan \alpha = -1 \rightarrow \alpha = \tan^{-1}(-1) = -\frac{\pi}{4}\]
anonymous
  • anonymous
Substitute this into either one of the sine or cosine expressions to get r:\[r \cos \alpha = r \cos (-\pi/4)=r \frac{1}{\sqrt{2}}=6 \sqrt{2} \rightarrow r = 12\]
anonymous
  • anonymous
what is happenin here
anonymous
  • anonymous
\[6\sqrt{2}\sin x -6\sqrt{2} \cos x = 12\sin(x - \frac{\pi}{4})\]
anonymous
  • anonymous
oh sweet, I think I can pick it up from here. Thanks so much!
anonymous
  • anonymous
No worries.
anonymous
  • anonymous
dam your goo at it
anonymous
  • anonymous
Your Wolframalpha answer is the same - \[12\sin(x - \pi/4) = 12 \sin (-(\pi/4 -x))=-12\sin (\pi/4-x)\]
anonymous
  • anonymous
thanks
anonymous
  • anonymous
oh, btw, on the step where you're dividing sin by cos to get tan, why did you divide 2 sins? Is that just a typo or am I missing something?
anonymous
  • anonymous
yeah, typo...
anonymous
  • anonymous
no prob, thanks a ton man, it makes sense now.
anonymous
  • anonymous
you're welcome

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