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April 12th, 2017, 09:22 PM  #1 
Member Joined: Nov 2016 From: USA Posts: 33 Thanks: 1  Direct method to solve for x in (cos x)/x=0.25
I can plug in numbers for x and then get closer & closer approximations. What is a quick, direct method to solve for x in: (cos x)/x = 0.25 
April 12th, 2017, 09:35 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,791 Thanks: 923 
There's no closed form answer. You have to find a numeric solution. You can use Newton's method $f(x) = \dfrac{\cos(x)}{x}0.25$ $f^\prime(x) = \dfrac{x\sin(x) \cos(x)}{x^2}$ $x_{n+1} = x_n  \dfrac{f(x_n)}{f^\prime(x_n)}$ and run this until it converges close enough for your satisfaction I get $x=1.25235$ Last edited by romsek; April 12th, 2017 at 10:21 PM. 
April 12th, 2017, 10:03 PM  #3 
Senior Member Joined: May 2016 From: USA Posts: 924 Thanks: 369 
Umm, for Newton's Method $f(x) = \dfrac{\cos(x)}{x}  0.25?$ Last edited by skipjack; April 13th, 2017 at 03:00 PM. 
April 13th, 2017, 10:19 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,978 Thanks: 807  Yes, Newton's method applies to equations of the form f(x) = 0. Here the problem was to solve $\dfrac{\cos(x)}{x}= 0.25$ which is the same as $\dfrac{\cos(x)}{x} 0.25= 0$.
Last edited by skipjack; April 13th, 2017 at 02:59 PM. 
April 13th, 2017, 04:09 PM  #5 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,545 Thanks: 518 Math Focus: Yet to find out.  Romsek missed the 0.25 in his original post, but edited it in after JeffM1 pointed it out .


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cos, direct, method, solve, x or x025 
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