Find number of solutions :

Question

dls · Answer

$$\Huge \sin \frac{\pi x}{2}=\frac{99x}{500}$$

tkhunny · Answer

Start with this, just for bounds:

$-1 \le \dfrac{99}{500}x \le 1$

$-\dfrac{500}{99} \le x \le \dfrac{500}{99}$

Second, what is the period of $\sin\left(\dfrac{\pi}{2}x\right)$?  It might seem reasonable to find two solutions in each period.  This might be an upper bound.

Give it some thought.  Let's see what you get.

tkhunny · Answer

The first inequalities are derived from an understanding of the sine function.  No matter what the argument, $-1 \le \sin(Whatever) \le 1$, as long as "Whatever" is valid.  It's just the way the sine function works.  You may think of it as "The Domain"?  What?  Yes, there was a reason why we studied that terminology and that concept.  The time is now.

tkhunny · Answer

First guess:  $\dfrac{\dfrac{500}{99}*2}{4} = 2.5ish $  Well, okay, if we get 25, we're probably worng.  10?  Probably wrong.  5 -- Hmmm...  Can't quite rule this out,

dls · Answer

its 7

tkhunny · Answer

How did you decide about the ones most distant from the Origin.  It was pretty close.

Did you work on both ends or did you observe they symmetry and do half the work?