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# Find the 3 smallest positive x-intercepts of the graph of and list them in increasing order.

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

Mar 20, 2020

#1
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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.

Mar 20, 2020
edited by Melody  Mar 20, 2020
#4
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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
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Well if it is just for fun there is definitely no need to ask many at once.

There is no point in asking questions if you do not learn from the answers.

Melody  Mar 20, 2020
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Why don't you try graphing it using Desmos graphing calculator?

Mar 20, 2020
#6
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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\\

Mar 22, 2020
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Hey rubikx2910,

Aren't you interested enough to comment on the answer.

Perhaps your assignment is past due so you have lost all interest ??

Melody  Mar 24, 2020