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August 27th, 2016, 04:22 PM  #1 
Newbie Joined: Aug 2016 From: Brazil Posts: 8 Thanks: 0  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 $\displaystyle \frac{20\pi}{3}$ Thank You Last edited by skipjack; August 28th, 2016 at 01:56 AM. 
August 28th, 2016, 12:33 AM  #2 
Newbie Joined: Dec 2006 Posts: 5 Thanks: 2 
Transform the sums in products using the formula $\displaystyle \cos a+\cos b=2\cos\frac{a+b}{2}\cos\frac{ab}{2}$ 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 $\displaystyle \cos 3a=4\cos^3a3\cos a$ and the equation becomes $\displaystyle 4\cos^3\frac{3x}{40}=0\Rightarrow \frac{3x}{40}=\frac{\pi}{2}\Rightarrow x=\frac{20\pi}{3}$ Last edited by skipjack; August 28th, 2016 at 01:58 AM. 

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