I have a Point P1 with the coordinates P(405747|5725660) in KBS EPSG:25832. Now I want to get the coordinates of a second Point P2. P2 is exactly 10m away from P1 and 30° from North.
Sorry for the bad drawing but I think it's clear what I mean. How do I get the coordinates of P2? For P(X2/Y2) is Y2 = Y1 + length * sin(Alpha). But what is the length? It can't be 10 can it? I feel that I need to convert the Meter into a different unit.
Not sure which Tags to use... but hope they are fine
If $P_1$ has coordinates $$ (x_1, y_1) $$ in metres (which is what https://epsg.io/25832 says are the units), and your drawing is correct so that the angle is measured clockwise from north (which I'm assuming is your $y$-axis, with the $x$-axis pointing to the right), and the distance is $10$ metres, then (because the distance is so small), we have $$ P_2 = (x_2, y_2) $$ where \begin{align} x_2 &= x_1 + 10 \cdot \sin(30)\\ y_2 &= y_1 + 10 \cdot \cos(30)\\ \end{align} where the sine and cosine are taken in degrees rather than the usual radians. Using 3-digit approximations for each of these, we get \begin{align} x_2 &= x_1 + 10 \cdot 0.500 = x_1 + 5.000\\ y_2 &= y_1 + 10 \cdot 0.866 = y_1 + 8.66\\ \end{align}
In your case, the starting point has $(x_1, y_1) = (405747, 5725660)$, so that $$ (x_2, y_2) = (4057\color{red}{52}, 572566\color{red}{8.66}) $$ where I've colored the changed digits in red.
Two things to note: 1. The earth's surface is curved, and for large distances, it's possible that the coordinate system's intrinsic distortions will cause the "everything is planar" approximation I've used here to be wrong. At a scale of 10 meters, however, it's surely correct to at least a couple of decimal points.
It's really easy to get things wrong in situations like this by not noting all your assumptions. I've assumed that
a. your starting point is measured in metres (because that's what the reference said),
b. that "up" on the page is the $+y$ axis,
c. that "right" on the page is the $+x$ axis,
d. that angles are measured clockwise from the $+y$ axis
e. that angle measurements are in degrees
The first item could be wrong if you're using a different reference than I am. The second and third "seem obvious", but they're not correct for the usual coordinate on a page of paper (where $x$ goes to the right and $y$ goes down, for instance, in Postscript). The fourth seems obvious from your drawing, but one should never assume, and the fifth is just a guess because you seem to be messing with navigation/geodesy/GIS stuff, where degrees are common.