“AAAAAAGGGGHHHHH!!!!!! TRIG MIDTERM TOMORROW!!!”
It seems that in every book or movie about high school students, there is always at least one kid who is horrified about an upcoming math test or homework assignment that brings about images of Armageddon. When a specific subject is mentioned, it always seems to be trigonometry (I’m going completely by memory; I haven’t taken a statistical sampling, but work with me here). It’s never calculus; I suppose because high school kids taking calculus are the ones who kick butt in math and are not inclined to complain about it. (Some of us actually liked it, but we were the ones you avoided and only show up in Hollywood films with annoying high-pitched nasal voices and horn-rimmed glasses held together with masking tape).
Incidentally, I never had trig as a stand-alone class, and I don’t know anybody who did. It was combined with second year algebra, which I think is fairly typical. I never once heard anybody in real life talk about a “trig exam” or “trig homework.” It was always “algebra” or simply “math.”
Anyway, trig does not deserve this bad wrap. Although “trig” is a 4-letter word, “trigonometry” clearly is not, and it need not inspire the spine-chilling horror that has made it the stuff of legend.
So what is trigonometry? As the first 3 letters imply, it is all about triangles. Specifically, it is about the relationships between the lengths of sides of a right triangle, which is a triangle that includes a right angle, i.e. a 90-degree angle, as shown in the picture below. (The square is standard notation for a right angle. I didn’t make it up.)
Because the right angle is a given for us to be thinking about any of this, it is not terribly interesting to consider further. The other angles are the interesting ones. We’ll consider the one on the left indicated by the Greek letter “theta” in the picture (we mathochists have always liked the Greek alphabet, go figure).
The basic building blocks of trigonometry are the sine, cosine, and tangent functions.
The sine (abbreviated “sin”) of the angle theta is defined as the ratio of the length of the side opposite to theta (labelled “b”) to the length of the hypotenuse (labelled “c”), which is the longest side of the triangle and is always opposite the right angle. In other words, it is b divided by c, or b/c in shorthand notation.
The cosine (abbreviated “cos”) of the angle theta is defined as the ratio of the length of the side adjacent to theta that is not the hypotenuse (labelled “a”) to the length of the hypotenuse, or a divided by c (a/c).
The tangent (abbreviated “tan”) of the angle theta is defined as the ratio of the length of the side opposite to theta divided by the length of the side adjacent, or b divided by a (b/a).
Everything in trigonometry comes from these three functions. And each one of them comes down to simple division, i.e. basic arithmetic. Q.E.D. This it is, and nothing more.
But so what, you might say. You mathochists can play around with your right triangles and your ratios all you want. What use do they have in the real world? Plenty.
A right triangle is a very stable and strong structure, and it is no accident that it is used as the basis for such structures as bridges. Below is a picture of a pedestrian/bike bridge I use many times a week in my morning bike rides. Lots of right triangles formed by the beams are keeping it together, for which I am grateful every time I ride my bike over it.
If you ever have the gumption to explore the crawl space between the ceiling and the roof of your house, you will see a lot of right triangles formed by the joists and beams holding your roof up. Structural integrity is a highly desirable quality in a roof, and right triangles do the job very well.
The people who design and build these structures routinely use these trigonometric relationships to determine the lengths of the different pieces that make up the structure, whether they are made of wood, aluminum, titanium, fiberglass, or anything else.
I use them every day in my work as an Electromagnetic Compatibility (EMC) engineer, but that is best left for another post.