How can the equation $\cos(3x) = \cos(x)$ not give the same answer as $\cos(x) = \cos(3x)$. I'm taking an analysis course and I think its important to have an intuitive feeling of whats going on but it doesn't make sense to me.
trigonometric equation with cos
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real-analysis
trigonometry
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0the have the same solutions because these are the same equation – 2017-01-26
2 Answers
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The equations $\ cos(3x) = \cos(x) $ and $\ cos(x) = \cos(3x) $ are the same.
We have $a=b$ iff $b=a$
1
$$ 4 c^3 - 3 c =c \, , \rightarrow c = 0,\pm 1 $$
Solutions are $ 0, m \pi/2, n \pi/2 $ for all integers $m,n$ no matter whichever order you write the equation:
$$ \cos x = \cos 3x,\, \cos 3 x = \cos x.\,$$