So I am working on a problem related to changing my initial direction, based on how strong and in which direction the wind blows. An illustration can be seen below
Assume I am located at $(0,0)$ and want to reach the point $(X,Y)$ decpicted by the red dot in the figure below. Assume that I am able to fly at a constant velocity of $v$ m/s while there is a wind blowing with an angle $\gamma$ relative to the horizontal plane, where the wind velocity is $w$ m/s. Both $\gamma$ and $w$ are given. The question how do I decide the new angle $\beta$ to fly with to get to point $(X,Y)$ taking the wind into consideration.
My initial thoughts was first to calculate $\alpha$ as
$$\alpha = \arctan \frac{X}{Y}$$
Similarly the distance it takes to reach $(X,Y)$ is
$$d = \sqrt{X^2 + Y^2}$$
Now I am stuck, any idea how the new angle $\beta$ should look?
First, decompose the velocity $v$ and $\omega$ over $x$-axis and $y$-axis. I have: $$X=(v_x+\omega_x)\cdot\Delta t$$ and $$Y'=(v_y+\omega_y)\cdot\Delta t=X\frac{v_y+\omega_y}{v_x+\omega_x}$$ From here, I can say that: $$\beta=\tanh\left(\frac{Y'}{X}\right)-\alpha=\tanh\left(\frac{v_y+\omega_y}{v_x+\omega_x}\right)-\alpha$$
You can easily compute $\omega_x=\omega\cos(\gamma)$, $\omega_y=\omega\sin(\gamma)$ and in the same way also $v_x$ and $v_y$.