Conversion of coordinates (longitude ; latitude) to (X;Y)

You have so many parentheses in your latitude formula that it’s hard to see what goes with what.

Let $\phi = \mathrm{LatY} \times \frac\pi{180},$ that is, if LatY is the latitude in degrees then $\phi$ is the latitude in radians. Let $h$ be mapHeight and let $w$ be mapWidth. Then your formula for $y$ becomes this in mathematical notation: $$ y = \frac h2 - \frac{w \ln\left(\tan\left(\frac\pi4 + \frac\phi2\right)\right) }{2 \pi} $$

This is similar to the formula found at http://mathworld.wolfram.com/MercatorProjection.html except for the scaling and translation factors (which you want in order to fit the output on the display).

Solving the equation for $\phi$ (still in radians), $$ \phi = 2 \arctan\left(\exp\left(\frac{2\pi}{w} \left(\frac h2 - y \right) \right)\right) - \frac\pi2. $$ Multiply by $\frac{180}{\pi}$ to get the answer in degrees.

Again it’s hard to be sure due to the profusion of parentheses, but the attempted formula seems to be equivalent to the mathematical equation

$$ \mathrm{lat} = \frac{\exp\left(-\frac{\left(Y - \frac h2 \right)}{w} \times 2 \pi \right) - \tan\left(\frac\pi4\right) \times 2}{\pi / 180}, $$

which is clearly quite different. The fact that $\tan\left(\frac\pi4\right)$ (which is just equal to $1$) occurs in there should be a red flag indicating that something was done in the wrong order.