The discussion revolves around approximating the nonzero root of the equation x² = sin(x) using the cubic Taylor polynomial approximation for sin(x). The original poster seeks to find the value of r = √15 - 3 as an approximation.
Participants are exploring the relationship between the Taylor polynomial approximation and the original equation. Some guidance has been provided regarding the approximation process, but there remains a focus on how to specifically obtain the value of r.
There is an emphasis on the approximation method and the treatment of the error term, which is assumed to be zero for the sake of simplification. The problem statement specifies that r is a nonzero root, which adds a layer of complexity to the discussion.
But how do I able to obtain the number r?jbunniii said:In order to get exact equality you need
$$\sin(x) = x - x^3/6 + E(x)$$
where ##E(x)## is the error due to using only two terms. As an approximation, we assume this error to be zero, so we pretend that
$$\sin(x) = x - x^3/6$$
and solve the equation
$$x^2 = x - x^3/6$$
As the problem statement says, ##r## is a nonzero root of the equation ##x^2 = \sin(x)##. You will find an approximation to this by solving the equation ##x^2 = x - x^3/6##.zjhok2004 said:But how do I able to obtain the number r?
Solved, thanks!jbunniii said:As the problem statement says, ##r## is a nonzero root of the equation ##x^2 = \sin(x)##. You will find an approximation to this by solving the equation ##x^2 = x - x^3/6##.