Is this trigonometric equation solvable?The equation is this: x sin ⁡ ( x ) = 90 − xI've tried to solve it myself, to no avail. After that, I went to a bunch of algebraic equations solvers, which gave me differing results. Is this equation solvable? By solving, I mean finding a definitive numerical answer for the variable x. If it is, then what is the answer (and steps), and if not, why not?

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Is this trigonometric equation solvable?The equation is this:  x  sin  ⁡  (  x  )  =  90  −  xI&#039;ve tried to solve it myself, to no avail. After that, I went to a bunch of algebraic equations solvers, which gave me differing results. Is this equation solvable? By solving, I mean finding a definitive numerical answer for the variable x. If it is, then what is the answer (and steps), and if not, why not?

Nkgopotsev1g · Accepted Answer

I am assuming that the angles are in degrees. Using Newton&#039;s method I get   x  =  50.7284012596462Here is what I did. First I wrote your equation in terms of radians as  f  (  x  )  =  (  180  ⋅  x  sin  ⁡  (  x  )  )      /    π  +  (  180  x  )      /    π  −  90  =  0Derivative of   f  (  x  ) is given by      f    ′    (  x  )  =  180  (  sin  ⁡  (  x  )  +  x  cos  ⁡  (  x  )  +  1  )Now define a new function  h  (  x  )  =  x  −  f  (  x  )      /        f    ′    (  x  )starting with   x  =  0, I repeatedly applied   h to get  h  (  0  )  =  1.570796326794897    h  (  1.570796326794897  )  =  0.78539816339745    h  (  0.78539816339745  )  =  0.88707380376403    h  (  0.88707380376403  )  =  0.88537798182847    h  (  0.88537798182847  )  =  0.88537762625368    h  (  0.88537762625368  )  =  0.88537762625367    h  (  0.88537762625367  )  =  0.88537762625367Note how the numbers change very little. I stopped when there was no visible change to get the answer in radians as   0.88537762625367 or in degrees  0.88537762625367  ∗  180      /    π  =      50.72840125964648    ∘  After I posted the reply I saw the OP&#039;s clarification that the angles are in radians. The same approach works with  f  (  x  )  =  x  sin  ⁡  (  x  )  +  x  −  90  ,         f    ′    (  x  )  =  1  +  sin  ⁡  x  +  x  cos  ⁡  x  ,     h  (  x  )  =  x  −  f  (  x  )      /        f    ′    (  x  )to get   x  =  91.11846207528532 as the final answer.

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