Question

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.

Answer 1

First, try pretending the 6x is a y, then get some primitive solutions and go from there

Answer 2

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?

Answer 3

yes, and what are those places.

Answer 4

but how is the 6x tied into the answer? like what does it mean in terms of how many places cos=1/2

Answer 5

there are only 2 places where cos is equal to 1/2 on the unit circle

Answer 6

Which are: \[\frac{5\pi}{3}, \frac{\pi}{3}\]

Answer 7

Yeah

Answer 8

So from there, divide both by 6, like so:\[\frac{\pi}{3}\times\frac{1}{6}=\frac[\pi}{18}\]

Answer 9

Sorry that should say: \[\frac{5\pi}{18}, \frac{\pi}{18}\]

Answer 10

both of those, when plugged in for x, will give you 0.5

Answer 11

However, your interval was (1,2pi] correct?

Answer 12

yes

Answer 13

@rizags

Answer 14

still confusing?

Answer 15

general solutions for y = cosx is\[x = 2n\pi \pm sin^{-1}y\]

Answer 16

in your case it is\[6x = 2n\pi \pm cos^{-1}(1/2)\]

Answer 17

\[x = \frac{2n\pi \pm \pi/3}{6}\]

Answer 18

now find solutions in (1,pi] sub set by putting n=0,1,2....

Answer 19

those will be solutions

Answer 20

So there imore than 2 solutions?

Answer 21

I am a little confused on your last comment.

Answer 22

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.

Answer 23

There was 12 solutions in all