Don't stop now! Trigonometry is one of the most useful branches of mathematics for a Shooter, Gunsmith or Machinist.
Right triangles are very useful items. They are composed of six parts, those being the three sides (hypotenuse, opposite side and adjacent side) plus the three included angles, one of which is the 90° right angle.
As an example, if we take Pythagoras' a bit further, we can use it to help us with range estimations on up hill and down hill shots, as well as with wind deflection.
For example, you are trying to make an uphill shot at a distant target and you have used your LASER range finder and it says you are 300 yards from the target. Do you hold for 300 yards and fire? If you do, you miss, shooting OVER the target because the actual distance the bullet must travel is less than the 300 yards the range finder gave you.
Why? Because 300 yard reading is the line of sight distance, and not how far the bullet is actually going to travel before striking the target, strange as that may sound!
If you know line of sight distance and you can figure out the angle up hill or downhill (using JF Comfort's improvised inclinometer!) you can use the formula: Sine of Angle A = the Opposite side divided by the Hypotenuse (S.O.H.) to find the vertical distance. Once you have that, you can then use the Pythagorean Theorem to find the true distance the bullet must travel.
In the drawing above, if your LASER range finder gave you 300 yards as the distance from you at Point A, to the target located at Point B, and using JF’s inclinometer, or a real one, you find that the angle between the ground at your feet and the target slopes upward at a 30° angle, you can then use your scientific calculator to figure out what the distance the bullet travels, which would be the line between you at Point A and Point C.
The Sine of 30° is .5 so we plug that into the equation to get: .5 = X/300
We can multiply both sides by 300 and figure out that X =150.
OK, so now we know the length of the side of the right triangle we call BC is 150. The Hypotenuse is 300. So we square 300 = 90000, and we subtract the square of 150 (22,500) from the 90000 to get 67,500. We take the square root of that find that the side of the ABC triangle known as AC is 259.8 yards long.
Call it 260. Hold for 260 and make the shot.
Same thing works for windage deflection.
The other two parts of the right triangle package that let you figure out what’s missing are:
C.O.A. = Cosine = Opposite Side divided by Hypotenuse, or a divided by c. In the triangle shown above, the opposite side is a, and the hypotenuse as always is side c, the longest side.
And:
T.O.A. = Tangent = Opposite Side divided by the Adjacent Side, or a/b.
You can remember the above ratios with the mnemonic “S.O.H-C.A.H.-T.O.A.) rhymes with Krakatau, the volcano that blew its top.
There are also three inverse ratios for finding the above as well.
Cosecant = Hypotenuse divided by Opposite Side
Secant= Hypotenuse divided by Adjacent Side
Cotangent= Adjacent Side divided by the Opposite Side
There are other applications for the above as well, such as figuring out how far it is across a river, or how tall a tree or mountain is, by constructing proportionate triangles.
But we’ll save that lecture for later, assuming anyone wants to know how to do it that is.