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How to find exact value of x/2=sinx?

  1. Nov 10, 2015 #1
    According to Wolfram it is approx. -1.895

    How do I find the exact value of the root though?
     
  2. jcsd
  3. Nov 10, 2015 #2
    What makes you think it is a root? It probably is transcendental.

    Of course, 0 is an exact solution, and +1.895 is also approximately a solution.
     
  4. Nov 14, 2015 #3
    so how do we find the exact value of 1.895?
     
  5. Nov 14, 2015 #4

    Mark44

    Staff: Mentor

    I don't believe there is a way to find the exact value of the root that is near either -1.895 or +1.895. You can approximate either of these roots to whatever precision is needed using any number of numerical techniques, such as Newton's Method, Newton-Raphson, bisection, and others.
     
  6. Nov 15, 2015 #5
    Thanks. So is it transcendental?
     
  7. Nov 15, 2015 #6

    Samy_A

    User Avatar
    Science Advisor
    Homework Helper

    I think it is transcendental as a consequence of the Lindemann–Weierstrass theorem.

    If ##sin(x)## is algebraic (##x\neq 0##), so are ##x## (being equal to ##2sin(x)##) and ##cos(x)##.
    That implies that ##e^{ix}## is algebraic too. But this contradict the Lindemann–Weierstrass theorem.
     
    Last edited: Nov 15, 2015
  8. Nov 15, 2015 #7

    OmCheeto

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    Gold Member
    2016 Award

    Wiki says the following:

    Numbers proven to be transcendental:
    ...
    sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem).
    ...
     
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