Thursday, December 3, 2020

Basic Trigonometry

David's article from a week ago, where he covered distance calculation using simple geometry, reminded me of a level of math that is no longer taught. My high school math class was the last year to be taught how to use a slide rule and the first to see the introduction of electronic calculators; I learned a different kind of math than what is taught now, and it relies on basic mathematical functions instead of digital processing. Some of those basics could come in handy when your batteries are dead and you need to crunch some numbers.

Warning: Math ahead, so take appropriate precautions.

David covered how to determine the distance to an object by using angles from two points a known distance apart and the tangent of one of those angles. Tangents are one of the three basic ratios used in trigonometry and are derived from a “right triangle”. A right triangle is a three-sided shape where one of the corners is a “right angle”, or 90 degrees.

Pythagoras, a Greek philosopher and mathematician that lived about 2500 years ago, figured out that right triangles were important and had unchanging properties. No matter their size, certain ratios and relationships between the sides and angles were always the same. The Pythagorean Theorem states that the sum of the squares of the two adjacent sides of a right triangle will equal the square of the side opposite the right angle. We often see this expressed as a^2 + b^2= c^2.
A quick way to make a right angle is to use this theory and use sides that are 3, 4, and 5 units (whatever units you want to use; feet, inches, meters, dollar bills, cans of beans, etc.) to make a triangle. 3^2=9, 4^2=16, 5^2=25 and if you put those numbers into the Theorem you do get a true statement that 9+16=25.
You now have a basic right triangle to build from, and the rest of plane geometry follows from that building block. This works well in the field when trying to lay out fence corners and other things that need to be square.

Let's stick with right triangles for a bit. Here's a picture of a generic right triangle with sides a, b, and c and angles A, B, and C, courtesy of Wikipedia. The 90 degree angle is commonly annotated with a small square at the angle, especially for hand-drawn triangles.

For our purposes, angle C is the right angle (90 degrees) and the side opposite it (c) is called the hypotenuse. Since all of the angles of a triangle have to add up to 180 degrees, A+B will always equal 90 degrees in a right triangle.

David's friend explained that the tangent of an angle is the ratio of the length of the side opposite that angle divided by the length of the side adjacent to it. That means that the tangent (tan) of angle A would be a/b. Expressed as tan A= a/b and using what used to be freshman math, you can convert that to a= tan A*b.

There are other useful constants that can be derived from a right triangle:
  • Sine (sin) of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. Sin A= a/c
  • Cosine (cos) of an angle is equal to the length of the side adjacent divided by the length of the hypotenuse. Cos A= b/c
  • Tangent can also be expressed or found by dividing the sin by the cos (tan=sin/cos) of that angle.
(Editor's Note: You can remember these using the mnemonic SOH CAH TOA. 
  • Sine = Opposite * Hypotenuse
  • Cosine = Adjacent * Hypotenuse
  • Tangent = Opposite * Adjacent
The Hypotenuse doesn't change, but the Opposite and Adjacent sides are always relative to the angle from which you are measuring. -- Erin)
Once you know one angle and either one of the trigonometric ratios or the length of one side, you can apply these formulas to determine all of the other parts of the triangle. 

Sin, cos, and tan tables used to be included in the backs of most math text books and every reference book had them. Digital calculators have made those tables obsolete, but they still exist and I'd recommend having a set somewhere in your library. If you're really lucky, you might be able to find an old slide rule that has the trigonometric functions on the static sides, handy for quick approximations.

I know that some eyes are glazed over by now, but these are basics of geometry which is the building block of engineering, architecture, and a lot of mathematics. Those are things you're going to need to understand if you want to do more than “eyeball” any construction or repair project. Think sloped roofs, fencing in odd-shaped areas, determining where a tree is going to fall, or how much cable you'll need to stabilize that radio antenna. Back when I worked around military imagery interpreters, they used this kind of math to determine the dimensions of buildings by measuring their shadows and the angle of the sun. Cartography (map making) uses this type of math extensively.

Next week I'll see if I can really bore you and try to explain logarithms and how they were used before the invention of computers.

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