Is my airspeed calculation code correct?

Question

Not sure if I should ask here, on Drones SE, or on Code Review SE.

Is my Python code for calculating airspeed given ground speed, ground direction, wind speed, and direction from which the wind is blowing correct? It takes directions in degrees, and the speeds can be in any units as long as they're the same; the output will be in the same unit.

import math


def find_airspeed(ground_speed, ground_direction, wind_speed, wind_direction):
    angle = math.radians(wind_direction - ground_direction)
     
    return math.sqrt(
        (ground_speed + math.cos(angle) * wind_speed) ** 2
        + (math.sin(angle) * wind_speed) ** 2
    )

Or, expressed as formulae:

$$\theta = (\Theta_{Wind} - \Theta_{Gnd})$$

$$AirSpeed = \sqrt{(V_{Gnd} + V_{Wind}\cdot cos(\theta)) ^2 + (V_{Wind} \cdot sin(\theta) )^2}$$

Answer 1

If you apply the law of cosines to the triangle formed by the vectors $\vec{A}$ (Airspeed), $\vec{W}$ (Wind) and $\vec{G_S}$ (Ground speed), where $\vec{G_S}=\vec{A}+\vec{W}$, you get:

$$|\vec{A}|=\sqrt{|\vec{G_S}|^2+|\vec{W}|^2-2|\vec{G_S}|\cdot|\vec{W}|\cdot\mathrm{cos}(\theta)}\tag{1}$$

where $\theta=\theta_{G_S}-\theta_W$.

As you yourself pointed out in the comments below, wind directions are usually reported as the direction winds originate from, not where the blow to. With that convention, the angle $\theta_W$ should be expressed as $\theta_W=\theta_W'+180$. It's easy to show that the additional $180$ can be brought out the cosine as a "-" minus sign. The final equation is therefore:

$$|\vec{A}|=\sqrt{|\vec{G_S}|^2+|\vec{W}|^2+2|\vec{G_S}|\cdot|\vec{W}|\cdot\mathrm{cos}(\theta)}\tag{2}$$

That's equivalent to yours.