Do pilots (for example commercial pilots) need to use trigonometry (directly, with trigonometry knowledge) as part of their routine job?
If so, when is it most used and for what?
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Note that this answer was given to the original question,
Do pilots need to use trigonometry as part of their routine job? If so, when is it most used and for what? The question was subsequently edited, after this answer was accepted. Much of the information provided in this answer does not now apply directly to the current question, but it does serve to provide context and for this reason it has been retained.
Pilots use trigonometry primarily for calculations involving wind angles, either for navigation or for cross-wind components for takeoff or landing. Note that pilots often accomplish these calculations with a flight computer such as the E6B, a circular slide rule:
Many such flight computers are available, including electronic versions, such as the Sporty's electronic E6B. Personally, I now prefer to use web-based software, but in flight school my preferred calculation method was a TI-83 programmed to do vector addition and other pre-flight calculations. In the cockpit, the CR-3—a pocket version of the larger E6B mechanical flight computer and which can be operated with one hand—is usually the most functional method.
Regardless of how they are performed, such calculations are typically for the purpose of either:
Calculating wind correction angles for cross-country navigation based on forecast wind conditions, or
Calculating crosswind component for takeoff or landing based on reported wind conditions.
The later are fairly straight-forward; the former are more complex. Crosswind component is a simple tangent calculation. Wind correction angles are more complex involving vector addition taking into account magnetic course, true airspeed, forecast wind angle, and forecast wind speed.
Now, the original question was merely, do pilots need to use trigonometry. The answer to that question is, "Yes". The question has since been edited to specify the use of trigonometry "directly, with trigonometry knowledge". The answer to that question is, "No", pilots do not need to have a working understanding of trigonometry, though they do need to be capable of computing the above described wind problems. A pilot need not understand the trigonometry used by the calculator, slide rule, or FMS that he or she uses to perform these calculations. I know good pilots who have no real working knowledge of trigonometry and do quite well without it.
For a pilot flying in the IFR environment, wind correction calculation is less frequently a needed calculation since most navigational means provide for wind correction in some way. This is especially true of GPS or RNAV navigation, because the ground track can be compared to the aircraft's heading to show wind correction angle.
I think the best example of routine use of trigonometry by pilots is to calculate winds aloft. The data a pilot has on hand is his compass heading and his airspeed and his altitude and his true bearing based upon passing landmarks on the ground. From this data he can calculate the wind direction and velocity he is witnessing at his location and altitude.
Normally he would use a flight computer called an E6B which is a circular slide rule of sorts. The E6B is easier to handle in turbulence than a pocket calculator. I proved this to myself many years ago when I added a program to a programmable pocket calculator to do the computation. It was hard to improve upon the time-tested E6B.