Trigonometry is infamously something most normal people fear and loath. Or at least don’t understand and don’t particularly want to deal with. (In fairness, it doesn’t pop up much in regular life.) As with matrix math, trig often remains opaque even for those who do have a basic grasp of other parts of math.
Excellent and thorough tutorials exist for those interested in digging into either topic, but (as with matrix math) I thought a high-altitude flyover might be helpful in pointing out important concepts.
The irony, as it turns out, is that trig is actually pretty easy!
At least the basics are. The thing to keep in mind about trigonometry is that it’s just triangles. And right-angle triangles at that.
[I’m reminded of an old calm down and carry on saying among computer geeks: “Calm down, it’s just ones and zeros.” So for trig: Calm down, it’s just triangles.]
But the other thing about trig (again, as with matrix math) is that it’s often taught by teachers without a fundamental understanding or who even share that vague fear and loathing. Math, in general, often suffers from rote teaching by teachers who learned it the same way.
If I may be forgiven for one last comparison to matrix math, in the defense of teachers these may not be the easiest subjects to teach. As with any subject, there are gestalts that provide a “big picture” view. Without those, one has little else to resort to than those rote rules and procedures which, at best, are boring and, at worst, are utterly opaque and off-putting.
And, to repeat, most people just don’t run into trigonometry very often in life. Who among us has ever needed to calculate the height of a flagpole using trigonometry? It does seem to fall under the frequent student complaint: Why must I learn this? What’s it good for?
The intent here is to try to provide those basic gestalts — a basic understanding of trigonometry on a fundamental level. And, perhaps, to demonstrate what it’s good for (besides figuring out how tall a flagpole is).
§
Figure 1. The Trig Triangle
As mentioned, it’s all just right-angle triangles.
Figure 1 shows the canonical trigonometry right-angle triangle. It stands for all triangles with the following two properties:
• One of the three corners has a 90° angle.
• The long side is set to a length of 1 (no matter how long it actually is).
Such triangles can be tall and skinny, or squat and broad, or (as in Figure 1) somewhere in the middle.
A primary characteristic of triangles in flat space is that the sum of their three angles is always 180°. (In fact, a test of a space to see if it’s flat is to see if this is true. If it isn’t, the space isn’t flat.) Since a trig triangle has a 90° angle, the other two angles must sum to 90°.
Effectively that means we only care about one of the angles (which we call theta, θ), since the other angle is always 90-θ. For example, if theta is 30° then the other angle is 90-30=60°.
Setting the long side (the hypotenuse) to 1 makes the math easier. For real-world triangles with real-world lengths, we use the actual value of the hypotenuse as a scaling factor. For example, if the hypotenuse is actually 4.5, then, as you’ll see, we scale our various trig values for the sides by that much.
Figure 2.
A key point is that the 90° angle along with locking the hypotenuse to 1 constrains the a and b sides such that, if the triangle is tall — side a is long — then side b is forced to be short, making the triangle skinny.
Conversely, if side b is long, side a is forced to be short, and the triangle is squat and broad.
Note that, as shown in Figure 2, the orientation of the triangle isn’t important, only that it has a 90° angle.
Note also that in Figure 2 I kept the hypotenuses the same length, even though that length (per the background grid) is obviously not 1. In fact the actual length (per the grid) is 4.1231… (it’s the square root of 17).
Which brings us to another important aspect of right-angle triangles and trigonometry: the Pythagorean theorem. It says the hypotenuse is the square root of the sum of the squares of the sides. Which always sounds torturous expressed in words. It’s much simpler in numbers:
Since we’re setting the hypotenuse to 1, we have:
If we take the square root of both sides:
Which, incidentally, is the equality that defines a circle (the unit circle, but as with the triangle, it can be scaled to any real-world size).
And now the locking of the side lengths is obvious, because:
Making one side longer makes the other side shorter, and vice versa. This proportional locking is fundamental to trigonometry — it’s why it works.
§
So trig boils down to one angle and one side of a right-angle triangle. How easy is that?
Not only are the two sides locked, so is the angle, theta, and this is where the trigonometry kicks in. Given that angle, and the length of one side, we can determine all the sides and angles of a right-angle triangle.
We can also determine the angle by knowing the lengths of at least two of the sides.
Essentially, given two pieces of information about the triangle, we can determine all the information about it. That’s the value of trig.
§ §
The trick is to not let words like sine and cosine (or arctangent!) intimidate. They’re attached to simple concepts.
[That said, as with most math topics, one can get deep into the weeds where things get very involved and there’s lots to learn. We won’t need any of that here.]
Figure 3. Sine & Cosine
To explain, consider Figure 3.
To be canonical, we need to switch from calling them side a and side b to calling them side y and side x, respectively. This places trigonometry on the Cartesian plane.
The sine and cosine are nothing more than the proportions of the sides given some angle, theta (θ). They are functions that take an angle and return the length of the side.
Note that the length returned assumes the hypotenuse is set to 1. As mentioned above, when that’s not true, the actual hypotenuse length becomes a scaling factor we multiply the returned sine or cosine value by to get the actual side length.
Again, all sine and cosine do is, given an angle, give us the length of a side assuming the hypotenuse is set to 1. That’s their whole deal.
In particular, the sine gives the length of the side opposite the angle, and the cosine gives the length of the side next to (“co”) the angle (see Figure 1).
§
You may have noticed that, while sine has its own Wiki page, cosine (and all other trig functions) link to the trigonometric functions page (or the inverse trigonometric functions page for ones like arctangent).
That’s because sine is the fundamental function. The cosine, due to how the sides are proportionally locked, is just sine’s mirror image. In fact, by simply switching to the other angle, the sine and cosine swap.
If you’ve ever perused a trig table, you’ve noted both contain the same numbers in reversed order. (Actually, not reversed but 90° out of phase with each other.)
[For the more mathematically inclined, the cosine is the derivative of the sine and vice versa, modulo some sign changes.]
At root, the sine is nothing more than the length of the opposite side divided by the length of the hypotenuse. Cosine is the length of the adjacent side divided by the hypotenuse. At heart, they’re just fractions that, because of the proportional locking, are tied to the angle. (See the Wiki trig functions page.)
A sine wave is what we get if we feed the sine function a progression of increasing angles:
Generating a red sine (and blue cosine) wave. [from Wikipedia]
The above animation nicely demonstrates how a sine wave comes from circular motion. In fact, the A.C. electricity that comes into our home follows a sine wave because generators turn in circles.
The animation also illustrates how the sine and cosine are 90° out of phase.
§
That’s pretty much the deal. The rest is just elaboration, details, and building on these basic tools.
Another big trig function is tangent, which is just the opposite side’s length divided by the adjacent side’s length. That ends up making the tangent the slope of the hypotenuse, which is why it isn’t defined for 90° (or 270°) — the slope is infinite on a vertical line.
[To be precise, the slope isn’t defined for a vertical line because slope is defined as Δy/Δx, and Δx is zero in a vertical line. The Δ (delta) just means change or difference.]
The other common trig function is arctangent, which is the inverse function to tangent. It takes the slope and returns the angle. (The inverse functions take a length and return the angle, whereas the normal functions take an angle and return a length.)
Arctangent is handy when you know the size of the triangle and want to figure out the angle. For example, the green triangle in Figure 2 has lengths of x=1.0 and y=4.0 and, thus, the slope of the hypotenuse is 4.0 (y/x).
My calculator says the arctangent of 4 is 75.9+ which means the angle (in the lower left) is nearly 76°. We can check this by using the angle and a known side:
Remember this assumes the hypotenuse is 1.0, but (per the background grid), that triangle’s actual hypotenuse is:
So that’s our scaling factor:
That vertical side is 4.0, so (given the rounding) exactly right.
I cannot stress enough that the only way to really learn these things is to work with them. Just reading about them won’t do it.
[As a side note to programmers, the common atan2 function provided by many code libraries is the arctangent function plus it does the slope calculation. The atan2 function takes two parameters, the two sides, calculates the slope and then returns the arctangent of that slope. It’s such a common operation the library just does it for you.]
§ §
This was a hard post to write. I’ve had a note about a “Trig Is So Easy” post for years, but I’ve been working with trig so long it’s hard to judge what needs to be explained and what is obvious.
And it just now occurs to me to wonder if that’s why Rachel Maddow often seems to over-explain things on her MSNBC show. She’s so smart and well-read that she misjudges what her viewers need explained and what is well-known to them? (One time she spent about five minutes explaining a “dead man switch.” I kind of assume anyone smart enough to watch Maddow in the first place already knows what that is, but maybe that’s my own blind spot speaking.)
Well, if that’s what’s going on there, I can sure relate. It ain’t easy!
Stay trigonometric, my friends! Go forth and spread beauty and light.
∇
Logos con carne 
February 18th, 2021 at 9:23 am
The height of flag poles aside, in physics trigonometry gives us a map between the angular location of a point and its x and y coordinates in space. It also gives us the distance (magnitude) of the point through the Pythagorean theorem.
If nothing else, associate cosine with the x coordinate and sine with the y coordinate.
There is also a strong connection with sine waves, and recall that any wave form is a combination of sine waves, so trig is also a tool for wave analysis.
February 18th, 2021 at 11:01 am
[sigh] Yet another reason to hate the WordPress Reader: It “grays out” older posts, a UI technique that usually means “disabled” but to the WP Reader apparently just means “old”.
Note you can still click into these posts and read them even though the graying out suggests you wouldn’t be able to. I’ve complained about this, but it’s apparently intended behavior.
Add that to how the Reader sometimes makes your paragraphs merge into one block of text if you use the Classic editor. If you use the Reader to comment, you might notice it does that to comments at first, but if you exit and come back, your paragraphs are usually intact.
The damn Reader also strips out style information, even if the (Classical) editor put it there in the first place. Did you center something? Well, it won’t be centered in the Reader. And forget about any font color or font style or indent you set, because it won’t be there in the Reader.
The only virtue I see in the Reader is that it also strips out ads for blogs that haven’t upgraded to having no ads (you’re welcome). There are a couple of blogs I follow that I won’t visit the actual site because of the ads, but I can read the posts ad-free in the Reader.
Kinda funny how the Reader also removes ad revenue. 😀
February 18th, 2021 at 11:09 am
Heh. I just noticed this is post #1100. 🙂
February 19th, 2021 at 10:49 pm
Saw this, thought of you: https://arstechnica.com/science/2021/02/study-blames-earths-magnetic-field-flip-for-climate-change-extinctions/
February 20th, 2021 at 2:52 pm
Because trigonometry? 😀 😀
March 14th, 2021 at 12:49 pm
[…] also Pi Day, so either do some trigonometry (it’s easy) or eat some pie. (But put me down for Tau Day when I ken has double the pizza […]
April 6th, 2021 at 12:11 am
I have never been good with numbers, all this homeschooling has me now throwing my hands in the air and making sad faces. Ill just slink off and feel…intimidated by triangles.
April 6th, 2021 at 8:26 am
It’s all about one’s early training, and for most that training hasn’t been good so it all does seem pretty intimidating. The hard part is getting started and facing those first foothills. Most just don’t see the need or point to it. A question I’ve heard a lot is “Why?!” If I can ever help, answer any questions, feel free to ask!
April 6th, 2021 at 10:37 am
Bit of proud mom moments – kiddo pulled his math grade up from a C to an A – I paid for homeschool…found him watching math lectures and up early slogging at it. He doesn’t bother asking me anymore. Thank you for the offer, Ill let him know if he gets stuck you were kind enough to offer! Math boggles me entirely.
April 6th, 2021 at 10:53 am
Well, you’re certainly not alone in that — a lot of people see it as a foreign language. Which is unfortunate and far more due to the teachers than the students. Anyone can learn it given a teacher who knows and loves the subject (although, sadly, learning anything gets harder as one ages — I’m trying to learn quantum mechanics, and it’s a challenge).
That said, I took German classes for four years in high school and it never took, like, at all, so I think people are just better at some things than others.
Kudos to the kiddo! Math fundamentals open a lot of doors, whether it be the sciences, baseball stats, or computer animation (which is where I started getting into trig). Learning math definitely falls under the heading of “teaching someone to fish.”
April 6th, 2021 at 10:43 am
…I went to a very nice school post 16. I was told to not bother attending math, just accept I would fail it. I got an A. which I am not wholly blaming on speed…but everything certainly made more sense for five minutes. As soon as I could, I dumped it and went where I belonged – liberal arts degree at a nice university. I used to stare at the stem guys in awe.
April 6th, 2021 at 11:01 am
Yeah, one of the really fucked up things is that whole “girls can’t do math” crap. Tell it to Emmy Noether or any of the other brilliant female mathematicians. Then watch out for the left hook…
I was a science nerd until I got into theatre in high school and discovered my artistic side, so I went from STEM to STEAM. My college years (as a film student) were firmly liberal arts, and one of my key phrases for what’s wrong with the world is: “The Death of a Liberal Arts Education” (or, really, just the death of a good education in general).
April 6th, 2021 at 11:14 am
The Boy has never been to conventional school. He might go for high school next year. He is testing a year ahead, and is steaming through the work. I am with you about the death of a good education. I’m old enough that I was still subject to a more rigorous schooling. I actually think I am ‘math dyslexic;’ – not the teacher’s fault. I had tutor after lovely tutor, I just can’t see numbers. It was, curiously, a girl’s only school that banned me from math. I had come from a non feepaying school and was so behind them that she was irritated by me. Film, huh! That sounds fun. English Lit here.
April 6th, 2021 at 12:45 pm
It does seem like our brains can be wired, or not wired, for certain things. I’m a retired software designer, and I’m fluent in over a dozen programming languages, but I just can’t seem to wrap my head around foreign languages. And I’ve met others who are the same when it comes to numbers. I get that way with my taxes and complicated forms in general. My mind just swims and drowns.
Theatre in high school, filmmaking in college. I lived in Los Angeles then, and I was gonna be the next George Lucas, but that thing about having to know someone in the industry was really true. (How do you get a Union Card so you can get work? You need 22 hours of work experience. How do you get those 22 hours? You need a Union Card. [D’Oh!]) I decided I just wasn’t willing to work my way up the ladder so maybe some day I might direct a film. Meanwhile, I fell into a job that spoke to my technical side, did well, and the rest, as they say, is history.
English Lit! Big reader? (From your blog it looks like you have some interest in science fiction.) I’m currently trying to read everything Agatha Christie wrote. My reading tastes are decidedly low-brow; mostly science fiction with a side of detective stories and murder mysteries. I’m a long-time fan of Christie’s Hercule Poirot, but only recently have I figured out what an amazing writer she was. I’m working my way through the Miss Marple series at the moment.
April 6th, 2021 at 1:13 pm
I used to be a reader, but the degree destroyed it. Now my reading interests are niche – sci fi, the Beat poets, non fiction, Japanese literature. Im currently reading Basho, which is more grist to the writing mill. I am tending towards writing at this point in life, over reading. I love Christie! I was just discussing with the Boy how there needs to be a really great Sherlock Holmes game for playstation. THey totally wasted the ones they did do, total borefests.
So you are an effects guy? Im sorry you didn’t get to direct…but it sounds like you found your work fulfilling.
I’m just watching Concrete Cowboy. I loved Elba in The Wire, but am just not feeling him in this. The young man playing his son is great. I am going to battle through and hope that it speeds up! The lighting is awful, I can barely see anything….
April 6th, 2021 at 1:40 pm
I generally had a pretty good career, although I was never a great fit in Corporate America. The one upside was a pension, although things got bad enough late in my career that I retired early which meant a reduced pension. Social Security started for me this year, and that’s been a big help. (And Medicare, so I no longer have to buy medical insurance; that’s helped, too. Things were starting to look a bit dicey.)
I would have been far more a fit in theatre or film, but success in those is seriously iffy. I had (and still have) some interest in special effects, and my theatre activities were certainly as a backstage guy (lighting mostly).
I’m not familiar with Concrete Cowboy, although I do like Elba’s work. (I did very much enjoy The Wire.) Luther always sounded kind of interesting. Those British shows are often a lot less violent and dark than our American shows. (I’ve had a growing dislike of the casual violence of media ever since Luke blew up the Death Star.)
April 6th, 2021 at 1:52 pm
Luther is terrifying, not gory but scary as heck. Really recommend it if you get the time. It sounds like you had an interesting career. Spiderman you say. DC or Marvel?
April 6th, 2021 at 2:05 pm
With the exception of, despite many tries, never being successful in love, my life has much to be thankful for, and I’ve really enjoyed just about every minute. I do count myself very fortunate.
Spiderman and, as I mentioned, Superman and friends. As a kid it was mostly DC, but when I got older the darker aspects of Marvel held some attraction. Quite some time ago I got seriously into graphic novels (“gnovels” I call them), but that seems to have faded. I generally don’t care for the big budget movies. Too much casual destruction, fight scenes always seem to involve people hitting each other, and the stories really bother my logical “yeah, but” side. The ones I’ve liked (Deadpool, for instance) are the ones that don’t take themselves too seriously and have some fun. I especially like the deconstruction of the Deadpool movies.
April 6th, 2021 at 2:38 pm
Deadpool is cool, especially the deconstruction of the third wall. I am always disappointed by the movies.
Love is overrated. Good friends is where it is at!
April 6th, 2021 at 3:23 pm
That certainly has been my experience!
April 6th, 2021 at 6:22 pm
😃
April 6th, 2021 at 1:58 pm
It’s on Netflix. I really struggle to find things on netflix to watch and it is my only TV.
I just realized the Giants are playing the Padres, and I cant fucking watch it. I hate MLB.
April 6th, 2021 at 2:14 pm
Netflix, okay, I’ll add it to my queue (although it has a lot in front of it).
Netflix, let’s see… So much depends on taste… Things I’ve really enjoyed: The Good Place, Community, Russian Doll, Happy!, Love, Death & Robots, Black Mirror, Disenchantment, The IT Crowd, Lucifer, Unbreakable Kimmy Schmidt, Dear White People, Black AF, Grace and Frankie, The Kominsky Method, Medical Police, Space Force,… But obviously those are all just my tastes, and I have no idea what you may have seen already.
April 6th, 2021 at 2:40 pm
I really loved Russian Doll. Black Mirror is always fun, but they lost me on one of the episodes that I found too ‘edgy’. I have been enjoying Longmire…yeah..I know…but it is a really good modern western, with a passable fake Clint!
April 6th, 2021 at 3:24 pm
Ever seen Rango?
April 6th, 2021 at 6:21 pm
Rango was cute!
April 6th, 2021 at 2:16 pm
Do you know about the Netflix “special codes”? Ways to dig deeper into their catalog. Google for [netflix special codes]
April 6th, 2021 at 2:40 pm
! Really!
June 23rd, 2021 at 9:49 am
[…] — for example, the three corners pick out a unique plane and a unique circle in that plane. (Triangles also ground trigonometry.) Tripods allow a stable platform on uneven or rough […]
December 7th, 2021 at 12:30 pm
The other irony is that, while being good at math can make one seem less human in popular vision, doing math is something that is especially human. We invented the stuff.
As I’ve said many times, being bad at math should be viewed the same way we view being bad at reading or writing. Don’t forget: Math was the third R! It’s a key way you don’t get fooled again.