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).
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.
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.]
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:
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.