cutiecomittee123
  • cutiecomittee123
How many solutions does the equation cos(6x)=1/2 have on an interval of (1,2pi]? I just need a few pointers on this one.
Mathematics
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SOLVED
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jamiebookeater
  • jamiebookeater
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Rizags
  • Rizags
First, try pretending the 6x is a y, then get some primitive solutions and go from there
cutiecomittee123
  • cutiecomittee123
Okay I need more info because I know that cos is y on the unit circle when trying to find solutions. Is this basically just saying "How many places does cos=1/2 on the unit cirlce?
Rizags
  • Rizags
yes, and what are those places.

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cutiecomittee123
  • cutiecomittee123
but how is the 6x tied into the answer? like what does it mean in terms of how many places cos=1/2
cutiecomittee123
  • cutiecomittee123
there are only 2 places where cos is equal to 1/2 on the unit circle
Rizags
  • Rizags
Which are: \[\frac{5\pi}{3}, \frac{\pi}{3}\]
cutiecomittee123
  • cutiecomittee123
Yeah
Rizags
  • Rizags
So from there, divide both by 6, like so:\[\frac{\pi}{3}\times\frac{1}{6}=\frac[\pi}{18}\]
Rizags
  • Rizags
Sorry that should say: \[\frac{5\pi}{18}, \frac{\pi}{18}\]
Rizags
  • Rizags
both of those, when plugged in for x, will give you 0.5
Rizags
  • Rizags
However, your interval was (1,2pi] correct?
cutiecomittee123
  • cutiecomittee123
yes
cutiecomittee123
  • cutiecomittee123
@rizags
lochana
  • lochana
still confusing?
lochana
  • lochana
general solutions for y = cosx is\[x = 2n\pi \pm sin^{-1}y\]
lochana
  • lochana
in your case it is\[6x = 2n\pi \pm cos^{-1}(1/2)\]
lochana
  • lochana
\[x = \frac{2n\pi \pm \pi/3}{6}\]
lochana
  • lochana
now find solutions in (1,pi] sub set by putting n=0,1,2....
lochana
  • lochana
those will be solutions
cutiecomittee123
  • cutiecomittee123
So there imore than 2 solutions?
cutiecomittee123
  • cutiecomittee123
I am a little confused on your last comment.
lochana
  • lochana
I didn't calculate answers. so I can't say how many solutions will be there. you can substitute n with 0,1,2... and see answers are in the given set.
cutiecomittee123
  • cutiecomittee123
There was 12 solutions in all

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