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Problem:

Find the 3 smallest positive x-intercepts of the graph of

 

\[y = \cos(12x) + \cos(16x)\]

 

and list them in increasing order.

 

Enter your answer as a list of x-coordinates from least to greatest.

 Jul 11, 2022
 #1
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The intercepts are pi/14, pi/7, 3*pi/14.

 Jul 11, 2022
 #2
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This should help:

 

 Jul 11, 2022
 #3
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Use the identity

\(\displaystyle \cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos \left( \frac{A-B}{2}\right).\)

Then,

\(\displaystyle \cos(12x)+\cos(16x)=2\cos(14x)\cos(2x)\\ =0\\ \text{ when } \\ 14x = \pi/2 + m\pi \\ \text{or when } \\ 2x = \pi/2 +n\pi, \ \text{ m and n integers.}\)

Smallest will be when m = 0, then x = pi/28, etc.

 Jul 11, 2022

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