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August 27th, 2016, 04:22 PM   #1
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indicate least positive angle

Indicate the smallest positive angle that verifies the equation $\displaystyle \cos(\frac{x}{4}) + \cos(\frac{x}{5}) + 3\cos(\frac{x}{10}) + 3\cos\frac{x}{20}=0$

ANSWER $\displaystyle \frac{20\pi}{3}$
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Last edited by skipjack; August 28th, 2016 at 01:56 AM.
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August 28th, 2016, 12:33 AM   #2
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Transform the sums in products using the formula
$\displaystyle \cos a+\cos b=2\cos\frac{a+b}{2}\cos\frac{a-b}{2}$
and we have
$\displaystyle 2\cos\frac{9x}{40}\cos\frac{x}{40}+6\cos\frac{3x}{ 40}\cos\frac{x}{40}=0$
$\displaystyle 2\cos\frac{x}{40}\left(\cos\frac{9x}{40}+ 3\cos\frac{3x}{40}\right)=0$
Then $\displaystyle \cos\frac{x}{40}=0\Rightarrow x=20\pi$
or $\displaystyle \cos\frac{9x}{40}+3\cos\frac{3x}{40}=0$
Now we use the formula $\displaystyle \cos 3a=4\cos^3a-3\cos a$
and the equation becomes $\displaystyle 4\cos^3\frac{3x}{40}=0\Rightarrow \frac{3x}{40}=\frac{\pi}{2}\Rightarrow x=\frac{20\pi}{3}$
Thanks from Nathalia

Last edited by skipjack; August 28th, 2016 at 01:58 AM.
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