jmartinez638
  • jmartinez638
How do you find all the solutions for 'sin(x/2 - pi/4) = sqrt2/2'?
Algebra
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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phi
  • phi
solutions imply an equation equations have an = sign your expression does not. is that a typo?
jmartinez638
  • jmartinez638
Yes, So sorry = sqrt2/2
jmartinez638
  • jmartinez638
Edited original.

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phi
  • phi
do you know how to "undo" sin ?
jmartinez638
  • jmartinez638
Like using identities?
phi
  • phi
more like, do \(\sin^{-1} \) to both sides
phi
  • phi
\[ \sin^{-1} \left(\sin\left(\frac{x}{2} - \frac{\pi}{4}\right) \right)=\sin^{-1} \frac{\sqrt2}{2} \]
phi
  • phi
on the left side, the inverse sin of the sin undoes the sin we are left with \[ \frac{x}{2} - \frac{\pi}{4}=\sin^{-1} \frac{\sqrt2}{2} \]
phi
  • phi
on the right side it is asking for an angle what angle is it where sin of that angle = sqr(2)/2 we want it in radians and it is an angle people memorize (so you should too)
jmartinez638
  • jmartinez638
45 degrees or \[\pi\]/4
phi
  • phi
yes. but because the question asks for *all* solutions we should eyeball the graph for sin |dw:1440367395779:dw|
phi
  • phi
so one family of solutions is pi/4 + 2pi n and the other is 3pi/4 + 2pi n where n is any integer
jmartinez638
  • jmartinez638
Oh that makes sense. In relation to the graph especially...
phi
  • phi
so we should try to solve \[ \frac{x}{2} - \frac{\pi}{4}=\frac{\pi}{4} + 2\pi n\] and also \[ \frac{x}{2} - \frac{\pi}{4}=\frac{3\pi}{4} + 2\pi n\]
jmartinez638
  • jmartinez638
\[(8n+5)\pi \div2\] for the first one
jmartinez638
  • jmartinez638
Is that a viable answer?
phi
  • phi
\[ \frac{x}{2} - \frac{\pi}{4}=\frac{\pi}{4} + 2\pi n \\ \frac{x}{2} = \frac{\pi}{2} + 2\pi n \\ x= \pi + 4 \pi n \]
jmartinez638
  • jmartinez638
Oops yeah, I must've looked at the second one and not the first :3 Let me try the first...
jmartinez638
  • jmartinez638
I got x=(4n+1)*pi
phi
  • phi
yes , that looks good
phi
  • phi
and for the 2nd equation?
jmartinez638
  • jmartinez638
Does (8n+5)π÷2 work?
phi
  • phi
I don't see how.
jmartinez638
  • jmartinez638
Ah, I seem to have made a mistake. How about \[x=2(2n+1)\pi\]?
phi
  • phi
yes, that looks good. though people would probably write it as 2pi(2n+1)
phi
  • phi
and of course we have to say n is any integer
jmartinez638
  • jmartinez638
Ok, of course. So those two equations, with 'n' designated as any integer, would be the solutions to that equation?
phi
  • phi
yes
jmartinez638
  • jmartinez638
That makes a lot of sense, thank you so much!
phi
  • phi
yw

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