Finding roots: cosine function of x

happyparticle
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Homework Statement
Finding roots cosine function of x
Relevant Equations
##\frac{-5}{4} +2cos(\frac{\pi d x}{Lv}) + cos(\frac{2\pi dx}{Lv}) = 0##
I need to find the zeros of this function where d,L,v are constants.
After several calculations I faced this equation.
I tried everything I know, but I can't solve this. Maybe I'm missing something or I must made a mistake earlier in the problem.
Thus, I would like to know if it is possible to find the roots analytically (wolfram gave me the roots).##\frac{-5}{4} +2cos(\frac{\pi d x}{Lv}) + cos(\frac{2\pi dx}{Lv}) = 0##
 
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By math rule of
cos 2kx= 2cos^2 kx-1
you can get quadratic equation of cos kx where ##k=\pi d/Lv##
 
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Brilliant!
However, since this is a cosine function I have the same answer for different x. I found that patern by guessing with the calculator. however, is there a way to find it mathematically.
I hope this is clear.
 
happyparticle said:
However, since this is a cosine function I have the same answer for different x.
Please show us what you get actually for further investigation.
 
happyparticle said:
Brilliant!
However, since this is a cosine function I have the same answer for different x. I found that patern by guessing with the calculator. however, is there a way to find it mathematically.
I hope this is clear.
I think you missed his point. You can use @anuttarasammyak's equation to substitute and analytically solve a quadratic equation.
 
I good example is ##cos(kx) + cos^2(kx) = -1/4##
##kx = 2.09##
However, ##kx = 4.18, 8.36, 10.45, ...## works aswell.
 
Could you show "cos kx = .. " for your original OP and the new example in #6 by solving quadratic equations ?
 
For example in #6, I have ##cos(kx) = \frac{-1 \pm \sqrt{1-1}}{2}##
##kx = arc cos( \frac{-1 \pm \sqrt{1-1}}{2})##
 
1-1=0 so you can delete root. Arccos(-1/2) is easy to get.
 
  • #10
happyparticle said:
Brilliant!
However, since this is a cosine function I have the same answer for different x. I found that patern by guessing with the calculator. however, is there a way to find it mathematically.
I hope this is clear.

Remember that |\cos kx | \leq 1, so any solution of the quadratic equation which is outside this range can be discarded.
 
  • #11
anuttarasammyak said:
1-1=0 so you can delete root. Arccos(-1/2) is easy to get.

I mean, I still have the same question since arc cos(-1/2) ##\approx## 2.09.
However If I plug kx = 4.18 in #6, it works and |cos 4.18| < 1.

Is my question clear or I just don't see your point?
 
  • #12
happyparticle said:
I mean, I still have the same question since arc cos(-1/2) ≈ 2.09.

Draw a unit circle and the angle in it. You get precise radian or degree. That's what your teacher expects you to do, I suppose.
 
  • #13
I understand this point. Where ##x = {2\pi /3, 4\pi/3, 2\pi/3 + 2\pi + ...}##
however, I'm looking to have an expression for all the possible values of x.
 
  • #14
$$\cos(\theta) = \cos(\theta + 2 n \pi), \quad \quad n \in \mathbb{Z}$$
 
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