There was a little trick I used to use for calculating the amount of cross-wind component when landing an aircraft with a wind, say 45 degrees off the nose. It also used sines, but of course sine-tables and calculators are of limited use when flying an aeroplane!
It worked thus, and all you need is an analogue wrist-watch or just an imagined clockface.
Angular difference of runway to wind's origin. Lets say 45 degrees. You think of that as 45 minutes on your watch and convert that to the proportion of an hour, so in this case 3/4 of the watchface. If the wind is 20 knots, then you have 15 knots (3/4 of 20) cross-wind component, ie the value of wind. pushing the aircraft sideways across the runway. We needed to know this as there were limits on how much crosswind a particular aircraft type could be landed with, before control was lost as it landed and slowed. Which could create disagreeable amounts of paper-work post-incident!
If the angular difference is say 30 degrees, then that's half the watch-face, therefore 1/2 the total windspeed, so if 18 knots, that's 9 knots cross-wind.
This technique is surprisingly accurate for sine values and is a really useful fast and dirty way of calculating the sorts of sums your method requires, whilst you're doing other things and can't be mucking about with calculators etc. With a little practice you can do such sum in your head.
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