General Solutions (1 Viewer)

nrumble42 · Apr 30, 2017

Find the general solutions to:

a). cos4x +cos2x = 1
b). cos3x= cosx

any help would be good !

InteGrand · Apr 30, 2017

nrumble42 said:
Find the general solutions to:

a). cos4x +cos2x = 1
b). cos3x= cosx

any help would be good !

$\noindent b) The solution is $3x = x + 2n\pi$ for some integer $n$ or $3x = -x + 2n\pi$ for some integer $n$. That is, $x = n \pi$ for some integer $n$ or $x = n\frac{\pi}{2}$ for some integer $n$. As $n$ varies through the integers, we observe that the solutions picked up are precisely multiples of $\frac{\pi}{2}$. This is because the ``$x = n\pi$'' solutions are all multiples of $\frac{\pi}{2}$ $\Big{(}$so these form a subset of the ``$x = n\frac{\pi}{2}$'' solutions$\Big{)}$, and we get all multiples of $\frac{\pi}{2}$ from the other part of the solution. Hence the general solution is $x = n\frac{\pi}{2}$ for some integer $n$.$

InteGrand · Apr 30, 2017

nrumble42 said:
Find the general solutions to:

a). cos4x +cos2x = 1
b). cos3x= cosx

any help would be good !

$\noindent a) To solve this, we can use a double angle formula. We know $\cos 4x = 2\cos^{2}2x - 1$. Substituting this into the original equation will give us a quadratic equation in $\cos 2x$, and you should be able to go from there.$

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General Solutions (1 Viewer)

nrumble42

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InteGrand

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InteGrand

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