How do you solve this?
The horizontal line will create a segment; go here:
Circle Sector and Segment
Not sure if this is the simplest way but you require half the area of a semi circle to be $\displaystyle \frac{1}{4}\times \pi\times r^2$ to equal the segment part of $\displaystyle \frac{1}{2}\times (\theta -\sin \theta) r^2$ giving
$\displaystyle \frac{1}{4}\times \pi\times r^2=\frac{1}{2}\times (\theta -\sin \theta) r^2$
$\displaystyle \frac{1}{4}\times \pi=\frac{1}{2}\times (\theta -\sin \theta) $
$\displaystyle \frac{\pi}{2}= \theta -\sin \theta $
Incidentally, there is a similar question in the Trigonometry forum, but it does not offer a precise solution (yet).What is the most simple way of going from this equation (i.e. what is the easiest way of solving it, obtaining an accurate, numerical value)?
It is not clear to me what the phrase "find... the line" means in the original question. It is possible that the distance of the bisecting line from the diameter is a nice number expression built from natural numbers, $\displaystyle \pi$, the four arithmetic operations, radicals, and trigonometric functions. However, there may not be such an expression since the number of reals is vastly greater than the number of expressions (even though both are infinite). So, sometimes the description "the number $\displaystyle x$ such that $\displaystyle x-\sin x=\pi/2$" is as good as one can get. I would ask the instructor to clarify what kind of answer is expected. I believe that in this case it is not something like asking for a hint.
The answer expected is the point down the distance of the radius of the circle, through which it would be possible to draw a horizontal line, in order to cut the semicircle into 2 parts exactly. What is the most simple way of obtaining an accurate numerical value for this point?
You have been given the required equation and shown where it comes from. You have been told to take a numerical approach in solving the equation (because an exact solution is not possible). As far as I can see, the question you originally asked has been completely answered.
It is up to you now to decide what to do. What numerical approaches have you been taught? Do you know how to use technology (such as a graphics calculator) to solve an equation? These issues (that were not flagged in the original post) do not belong in the Geometry subforum.