I need some help with this problem.
I have a GPS coordinate and an angle in degrees. I need a new GPS coordinate x km away from the point I already have.
Degree is counted clockwise and y-axis is pointing at north.
Anybody that can help me out with a formula?
Except near the poles, for a distance as short as 5 km, one can treat the coordinate system as orthogonal, though not orthonormal (to stretch the meaning of these terms a bit). Denote the known point's coordinates by $\phi$ and $\lambda$. Then the unknown point's coordinates (taking north and east as positive) are
$$ \phi' = \phi - \frac{900 \sin 55}{\pi R \cos \lambda} $$ $$ \lambda' = \lambda - \frac{900 \cos 55}{\pi R} $$
where $R \doteq 6370$ is the Earth's radius in km. The $900/\pi R$ comes from $5/R$ generating the subtended angle in radians, multiplied by $180/\pi$ to get degrees. So, for instance, if one starts at $\phi = 2, \lambda = 49$ (near Paris), and travels $5$ km at heading $235$ (as in the diagram you provided), one ends up at
$$ \phi' \doteq 2 - 0.05615 = 1.94385 $$ $$ \lambda' \doteq 49 - 0.02580 = 48.97420 $$
Is that good enough? Near the poles it's a bit more involved.