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April 12th, 2017, 09:22 PM   #1
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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
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April 12th, 2017, 09:35 PM   #2
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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 $x=1.25235$
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Last edited by romsek; April 12th, 2017 at 10:21 PM.
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April 12th, 2017, 10:03 PM   #3
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Umm, for Newton's Method

$f(x) = \dfrac{\cos(x)}{x} - 0.25?$
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Last edited by skipjack; April 13th, 2017 at 03:00 PM.
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April 13th, 2017, 10:19 AM   #4
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Quote:
Originally Posted by JeffM1 View Post
Umm, for Newton's Method

$f(x) = \dfrac{\cos(x)}{x} - 0.25?$
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.
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April 13th, 2017, 04:09 PM   #5
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Originally Posted by Country Boy View Post
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 $\dfrac{\cos(x)}{x}- 0.25= 0$.
Romsek missed the 0.25 in his original post, but edited it in after JeffM1 pointed it out .
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