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.
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!
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\\