Find the 3 smallest positive x-intercepts of the graph of \(y = \cos(12x) + \cos(14x)\)and list them in increasing order.

Thank you in advance!

rubikx2910 Mar 20, 2020

#1**-1 **

What do you think you are doing?

14 question almost in a row.

We are not here to do all your homework or assignment for you!

I have hidden most of your questions.

Maybe I will slowly unhide them over the following days.

That is if you show any indication that you are learning from your answerers.

Melody Mar 20, 2020

#4**+1 **

Oh, it's not homework for my school or classes, I just found them in a book I had which I didn't have answers for. Sorry about that!

rubikx2910
Mar 20, 2020

#6**+1 **

You can use the identity

\(\boxed{cos(a)+cos(b)=2\;cos\frac{a+b}{2}\;cos\frac{a-b}{2}}\\~\\\)

\(y = \cos(12x) + \cos(14x)\\ \text{roots are given by}\\ \cos(12x) + \cos(14x)=0\\ 2\cos\frac{14x+12x}{2}\;\cos \frac{14x-12x}{2}=0\\ 2\cos(13x)\;\cos (x)=0\\ \cos(13x)=0\qquad \qquad or \qquad cos(x)=0\\ 13x=\pm \frac{\pi}{2}+2\pi n \qquad or \qquad x=\pm\frac{\pi}{2}+2\pi n\\ x=\pm \frac{\pi}{26}+\frac{2\pi n}{13} \qquad or \qquad x=\pm\frac{\pi}{2}+2\pi n\\ x=\pm \frac{\pi}{26}+\frac{4\pi n}{26} \qquad or \qquad x=\pm\frac{13\pi}{26}+2\pi n\\ x=\frac{\pi}{26}(4n\pm1) \qquad or \qquad x=\pm\frac{13\pi}{26}+2\pi n\\ \)

n | 0 | 1 | 1 | 2 | 2 | 3 | 3 | |

x | \(\frac{\pi}{26}\) | \(\frac{3\pi}{26}\) | \(\frac{5\pi}{26}\) | \(\frac{7\pi}{26}\) | \(\frac{9\pi}{26}\) | \(\frac{11\pi}{26}\) | etc |

Coding:

y = \cos(12x) + \cos(14x)\\

\text{roots are given by}\\

\cos(12x) + \cos(14x)=0\\

2\cos\frac{14x+12x}{2}\;\cos \frac{14x-12x}{2}=0\\

2\cos(13x)\;\cos (x)=0\\

\cos(13x)=0\qquad \qquad or \qquad cos(x)=0\\

13x=\pm \frac{\pi}{2}+2\pi n \qquad or \qquad x=\pm\frac{\pi}{2}+2\pi n\\

x=\pm \frac{\pi}{26}+\frac{2\pi n}{13} \qquad or \qquad x=\pm\frac{\pi}{2}+2\pi n\\

x=\pm \frac{\pi}{26}+\frac{4\pi n}{26} \qquad or \qquad x=\pm\frac{13\pi}{26}+2\pi n\\

x=\frac{\pi}{26}(4n\pm1) \qquad or \qquad x=\pm\frac{13\pi}{26}+2\pi n\\

Melody Mar 22, 2020