Trigonometric Equation with Unequal Coefficients: How to Solve?

[John O' Meara]

Solve $$3\cos\frac{3x}{2}=\cos\frac{x}{2}$$ I am interested to know how to go about solving such an equation where the coefficient of cos(x) is not equal to 1. I know how to solve $$\cos\theta = \cos\alpha, \mbox{ it is just} \theta = 2n\pi +/-\alpha$$ I teach various maths subject to myself, then I realized I didn't know how to solve the above equation. Thanks.

 

[l'Hôpital]

Well, $$\cos \frac{3x}{2} = \cos (x + \frac{x}{2})$$

Therefore, you get $$x = (2n + 1)\pi \ for \ n = ...-3, -2, -1,0,1,2,3,...$$ as a freebie.

 

[Gerenuk]

Surely the non-trivial solutions are more interesting.

You can always transform multiple angles to powers of trigonometric functions (and vice versa).
For example with de Moivre
$$\cos ny=\Re\left(\cos y+i\sin y\right)^n$$
$$\cos 3y=\cos^3y-3\cos y\sin^2y=\cos^3y-3\cosy(1-\cos^2y)=\cos y(4\cos^2y-3)=\cos y(2\cos 2y-1)$$

Since here on angle is three times the other, you can use y=x/2 and get
$$3\cos 3y=\cos y$$
$$3\cos y(2\cos(2y)-1)=\cos y$$
and therefore either
$$x=\pm\frac{\pi}{4}+k\pi$$
or
$$x=\pm\frac14\arccos\frac23+k\frac{\pi}{2}$$

 

[John O' Meara]

Thanks for the time you spent on working that out. I didn't know of De Moivre's. I worked it out afterwards and get $$2y=\pm \arccos(\frac{2}{3})+ 2n\pi$$. And y=x/2, I couldn't $$x=\pm\frac{\pi}{4}+k\frac{\pi}{2}$$, which is what you got. According to my graphing calculator the graph crosses the x-axis at $$\arccos(\frac{2}{3})$$ which looks close to pi/4 but is not. It crosses the x-axis again close to pi.

 

[Gerenuk]

Not sure what you mean.

Both 1/4arccos 2/3 and pi/4 are solutions.

 

[John O' Meara]

I get a principle angle of arccos(2/3) = 2y and -arccos(2/3) is another one , right. Maybe I have already gone wrong. If I saw the steps you used I could follow you. $$3\cosy(2\cos(2y)-1)=cosy, \mbox{therefore}\\ 3(2\cos(2y)-1)=1, \\ 2\cos(2y)-1=\frac{1}{3}, \\ 2\cos(2y)=4/3, \\ \cos(2y)=\frac{2}{3}$$

 

[Gerenuk]

The equation is
$$3\cos y(2\cos(2y)-1)=\cos y$$
Now either cos(y) is zero (which gives you pi/4 for x) or you can divide by cos(y) and continue to get x=arccos(2/3)/4

 

[John O' Meara]

Thanks.

 

[willem2]

Since here on angle is three times the other, you can use y=x/2 and get
$$3\cos 3y=\cos y$$
$$3\cos y(2\cos(2y)-1)=\cos y$$
and therefore either
$$x=\pm\frac{\pi}{4}+k\pi$$
or
$$x=\pm\frac14\arccos\frac23+k\frac{\pi}{2}$$

If y = x/2 then x = 2y, so if the solutions for y are

$$y = \frac {\pi} {2} + k \pi$$

or

$$y = \pm \frac {1} {2} \arccos \frac {2}{3} + k \pi$$

then the solutions for x will be

$$x = (2k + 1) \pi$$

or

$$x = \pm\arccos \frac {2}{3} + 2 k \pi$$

 

[John O' Meara]

Yes, that is what I finally finished up with, willem2.Thanks willem2.