Simplifying trigonometric equationI am a software developer who has run into a wall with a math equation.I have everything boiled down to this one equation, but I cannot seem to solve for w.My known variables are r, x, and s, and I need to solve for w. w = sin ⁡ ( π ( r − x w ) 2 ( r − s ) ) ∗ 2 rAny help is appreciated.Thanks!

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Simplifying trigonometric equationI am a software developer who has run into a wall with a math equation.I have everything boiled down to this one equation, but I cannot seem to solve for w.My known variables are r, x, and s, and I need to solve for w.  w  =  sin  ⁡  (            π      (      r      −      x      w      )              2      (      r      −      s      )        )  ∗  2  rAny help is appreciated.Thanks!

embutiridsl · Accepted Answer

This is too long for a comment.Under some conditions, you could approximate the solution.Let me change notation and define  y  =            π      (      r      −      w      x      )              2      (      r      −      s      )          ⟹    w  =      r    x    −            2      (      r      −      s      )              π      x        yThe equation becomes      r    x    −            2      (      r      −      s      )              π      x        y  =  2  r  sin  ⁡  (  y  )For   0  ≤  y  ≤  π, a very crude approximation of mine is  sin  ⁡  (  y  )  ≈      120          π      5        (  π  −  y  )  ywhich would let you with a quadratic equation in   y.  240  r  x      y    2    +      (    −    240    π    r    x    −    2          π      4        r    +    2          π      4        s    )    y  +      π    5    r  =  0Much better would be, for the same range, the magnificent  sin  ⁡  (  y  )  ≃            16      (      π      −      y      )            y              5              π        2            −      4      (      π      −      y      )            y          (  0  ≤  y  ≤  π  )proposed, more than 1400 years ago, by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician . This will give you a cubic equation to solve.  8  (  s  −  r  )      y    3    +  4  π  (  r  (  8  x  +  3  )  −  2  s  )      y    2    +  2      π    2    (  5  s  −  r  (  16  x  +  7  )  )  y  +  5      π    3    r  =  0

Alice Chen · Accepted Answer

I don&#039;t think an analytic solution exists. You may do better by defining a function  f  (  w  )  =  w  −  2  r  sin  ⁡      (                  π        (        r        −        x        w        )                    2        (        r        −        s        )              )  and doing numerical root-finding to numerically find the value of w which drives your function to 0, using for example Bisection Method or Newton&#039;s Method (since derivative is easy to find analytically and implement as well).

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