Editing Apparent wind (section)

==Calculating apparent velocity and angle==
{{unreferenced|section|date=September 2016}}
<math display="block">A = \sqrt{W^2 + V^2 + 2WV\cos{\alpha}}</math>

Where:
* <math>V</math> = velocity (boat speed over ground, always ≥ 0)
* <math>W</math> = true wind velocity (always ≥ 0)
* <math>\alpha</math> = true pointing angle in degrees (0 = upwind, 180 = downwind)
* <math>A</math> = apparent wind velocity  (always ≥ 0)
The above formula is derived from the [[Law of cosines]] and using <math>\cos(\alpha') = \cos(180^\circ-\alpha) = -\cos(\alpha)</math>.

The angle of apparent wind (<math>\beta</math>) can be calculated from the measured velocity of the boat and wind using the inverse cosine in degrees (<math>\arccos</math>)

<math display="block"> \beta = \arccos \left( \frac{W\cos \alpha+V}{A} \right) = \arccos \left( \frac{W\cos \alpha+V}{\sqrt{W^2 + V^2 +2WV\cos{\alpha}}} \right)</math>

If the velocity of the boat and the velocity and the angle of the apparent wind are known, for instance from a [[Apparent wind#Instruments|measurement]], the true wind velocity and direction can be calculated with:

<math display="block">W = \sqrt{A^2 + V^2 - 2AV\cos{\beta}}</math>

and

<math display="block"> \alpha = \arccos \left( \frac{A\cos \beta-V}{W} \right) = \arccos \left( \frac{A\cos \beta-V}{\sqrt{A^2 + V^2 -2AV\cos{\beta}}} \right)</math>

''Note:'' Due to quadrant ambiguity, this equation for <math> \alpha </math> is only valid when the apparent winds are coming from the [[starboard]] direction (0° < ''β'' < 180°).  For [[port]] apparent winds (180° < ''β'' < 360° or 0° > ''β'' > -180°), the true pointing angle (''α'') has the opposite sign:

<math display="block"> \alpha = -\arccos \left( \frac{A\cos \beta-V}{W} \right) = -\arccos \left( \frac{A\cos \beta-V}{\sqrt{A^2 + V^2 -2AV\cos{\beta}}} \right)</math>