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

[greswd]

According to Wolfram it is approx. -1.895

How do I find the exact value of the root though?

 

[Khashishi]

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.

 

[greswd]

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.

so how do we find the exact value of 1.895?

 

[Mark44]

According to Wolfram it is approx. -1.895

How do I find the exact value of the root though?

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.

 

[greswd]

Thanks. So is it transcendental?

 

[Samy_A]

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.

 

[OmCheeto]

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

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).
...​