It's Just Comparing

You are standing at the bottom of a hill, looking up. How steep is it?

You already know the answer intuitively. A gentle slope barely rises — you walk a long way forward for every foot you gain in height. A steep slope is the opposite: a lot of rise, not much forward distance. But here is the interesting thing. Without realizing it, you just answered a question about steepness by comparing two lengths: how far up versus how far forward.

That comparison — one side of a triangle divided by another — is the entire subject of trigonometry.

I know. "Trigonometry" sounds like the kind of word designed to make people stop reading. It shows up in conversations as a stand-in for "something impossibly complicated." But let me ask you this: did comparing the rise of that hill to its forward distance feel complicated? Because that is what trigonometry is. Two sides of a triangle, compared.

The beautiful part is what happens next. It turns out that these comparisons — the ratio of one side to another — depend only on the angle of the triangle. Not on how big the triangle is. A small ramp and a massive mountain with the same incline give you the exact same ratio. That single fact connects triangle measurements to circles, circles to waves, and waves to the physical behavior of sound, light, and everything that oscillates.

Here is my promise: by the end of this page, you will understand why these ratios are locked to the angle, not just that they are. And you will watch a wave emerge from a spinning circle in real time — and see exactly why it had to.

Let's start with a triangle and a question that will carry us the rest of the way: what stays the same when you make a triangle bigger?

What Stays the Same #

In the geometry essay, you learned that similar triangles have the same angles and proportional sides. Here is a consequence of that fact that is easy to state but genuinely surprising the first time you watch it happen.

Draw a right triangle with one angle fixed at 37 degrees. Measure the side opposite that angle and the hypotenuse. Divide the first by the second. You get a number — roughly 0.602.

Now double every side of the triangle. The opposite side is twice as long. The hypotenuse is twice as long. But twice-something divided by twice-something-else is... the same number. 0.602.

Scale the triangle tenfold. The opposite side is ten times longer. So is the hypotenuse. The ratio? Still 0.602.

This is worth pausing on. The triangle got enormous. Every side changed. But the comparison between the opposite side and the hypotenuse did not move. The ratio is locked to the angle.

And it is not just that one ratio. There are three natural comparisons you can make between the sides of a right triangle:

• The opposite side compared to the hypotenuse
• The side next to the angle (the adjacent side) compared to the hypotenuse
• The opposite side compared to the adjacent side

All three ratios depend only on the angle. Change the size of the triangle, and all three hold steady. Change the angle, and all three shift in lockstep.

Try it yourself. In the interactive below, drag the corner of the triangle to resize it — make it as large or as small as you like. Watch the three ratio readouts. They do not budge. Then drag to change the angle, and watch all three ratios respond.

You have just discovered something that most students who memorize SOH-CAH-TOA never see for themselves: the ratio between two sides of a right triangle is determined entirely by the angle. Not by the triangle's size. Not by how you drew it. Just the angle.

Three ratios. Three comparisons. Each one locked to the angle and nothing else.

Now — what do we call these comparisons?

Three Names You Don't Need to Memorize #

We have found three ratios that depend only on the angle. Before I give them names, let me try something.

Look at the triangle below. Focus on that ratio of the opposite side to the hypotenuse — the one you just watched hold steady at 0.602 when the angle was 37 degrees. This ratio is a function of the angle. It takes in an angle and gives back a number.

If you were inventing mathematics from scratch and you needed a name for this function — "the ______ of the angle" — what word might you choose?

The name that stuck, centuries ago, is sine. The sine of an angle is the ratio of the opposite side to the hypotenuse. For our 37-degree triangle, $\sin(37°) \approx 0.602$.

The ratio of the adjacent side to the hypotenuse has a name too: cosine. Think of it as the companion to sine — the other comparison that involves the hypotenuse. For 37 degrees, $\cos(37°) \approx 0.799$.

And the ratio of the opposite side to the adjacent side is the tangent. For 37 degrees, $\tan(37°) \approx 0.754$.

Three comparisons. Three names. That is all SOH-CAH-TOA was trying to say:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \qquad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \qquad \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

But here is why the names matter less than the understanding: if you forget the word "sine" tomorrow, you can reconstruct it. You know there is a ratio of opposite to hypotenuse. You know it depends only on the angle. You know why it depends only on the angle — because similar triangles have proportional sides. The name is a convenience. The understanding is the thing.

Let me show you that this is not just abstract. Remember measuring with triangles in the geometry essay? When you compared a stick's height to its shadow length to figure out the sun's angle, you were computing a tangent. You did not need the word then, and you do not strictly need it now. But having a name for a concept you already understand — that is useful. That is what good notation does.1

$$\tan(\theta_{\text{sun}}) = \frac{\text{stick height}}{\text{shadow length}}$$

You now have names for three comparisons, and — more importantly — you understand why those comparisons work. They are consequences of similar triangles, nothing more.

But so far, every triangle we have drawn has had an arbitrary hypotenuse — whatever length happened to come from the drawing. What if we were deliberate about it? What if we built a triangle where the hypotenuse is exactly 1?

When the Hypotenuse Is 1 #

Here is a thought that seems almost too simple to be useful.

Take any right triangle with angle $\theta$. The sine of that angle is the opposite side divided by the hypotenuse:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$

Now imagine choosing a triangle where the hypotenuse is exactly 1. Then:

$$\sin(\theta) = \frac{\text{opposite}}{1} = \text{opposite}$$

The sine is the opposite side. Not the ratio of the opposite side to something else — just the opposite side itself. And cosine becomes:

$$\cos(\theta) = \frac{\text{adjacent}}{1} = \text{adjacent}$$

The cosine is the adjacent side. That simplification is worth its weight in gold, because it means the ratios become actual lengths you can see and measure.

But notice where that far end of the hypotenuse sits. If the hypotenuse is 1, the far end is at distance 1 from the origin. It sits on a circle of radius 1. Every possible angle gives you a different point on that circle. And the coordinates of that point are... the adjacent side and the opposite side. Which we just said are cosine and sine.

Every point on a circle of radius 1 sits at $(\cos\theta, \sin\theta)$.

This is not a formula to memorize. It is an observation. You built a right triangle inside a circle. The horizontal leg is the cosine. The vertical leg is the sine. The hypotenuse is the radius, which is 1. That is the entire unit circle — a circle with radius 1, where the coordinates of every point are the trig ratios of the angle.2

You now know the coordinates of every point on this circle. But something else is hiding here — and I want you to discover it for yourself.

The wave

Before you interact with the circle below, I want you to make a prediction.

As a point moves counterclockwise around the circle, its height changes. At the rightmost point (0 degrees), the height is 0. As the point climbs, the height rises. At the top (90 degrees), it reaches 1. Then it falls back toward 0 at the left side.

If you plotted that height against the angle — a graph with angle on the horizontal axis and height on the vertical — what shape would the curve be? A zigzag? A series of bumps? Something smooth? Something jagged? Hold that prediction in your mind.

Now drag the point around the circle and watch.

The height starts at 0, rises to 1 at the top, falls back to 0 at 180 degrees. But it does not stop. Below the horizontal axis, the height goes negative — the vertical leg of our triangle now points downward. The sine value tracks this: positive above the axis, negative below. We have quietly extended sine beyond the right-triangle definition, from a ratio of positive lengths to a signed coordinate on a circle. This extension is what makes the wave possible.3

The height drops to -1 at the bottom of the circle, then climbs back to 0 at 360 degrees.

Now look at the shape. You know that shape. You have seen it on oscilloscopes, in physics textbooks, in diagrams of sound waves. That endlessly repeating curve has a name. It is called a sine wave. And you just watched it come from a circle.

And when the point completes a full trip and starts around again, the wave repeats exactly. Same shape, same height, same timing. That repetition — the same curve cycling endlessly — is what makes it a wave.

The wave is the circle, unrolled.

This is where sine waves come from. Not from a formula. Not from a textbook definition. From the height of a point walking around a circle.

You have just seen something remarkable: a wave emerging from a circle. The sine wave is not an abstract function defined by a formula. It is the height of a point going around in a circle, plotted over time.

But there is something else hiding in that circle. Look at the triangle you drew inside it — the one with the horizontal leg, the vertical leg, and the hypotenuse of 1. Does it remind you of anything from the geometry essay?

An Old Friend in Disguise #

There is a famous identity in trigonometry:

$$\sin^2(\theta) + \cos^2(\theta) = 1$$

It shows up in every trig textbook. Students memorize it. It appears on formula sheets. And it is, I want to argue, the least surprising fact in all of trigonometry — once you see where it comes from.

Look at the unit circle again. At any angle $\theta$, you have a right triangle inside the circle. The horizontal leg has length $\cos(\theta)$. The vertical leg has length $\sin(\theta)$. The hypotenuse has length 1 — it is the radius.

Now apply the Pythagorean theorem to this triangle:

$$a^2 + b^2 = c^2$$ $$\cos^2(\theta) + \sin^2(\theta) = 1^2$$ $$\sin^2(\theta) + \cos^2(\theta) = 1$$

That is the entire derivation. The "Pythagorean identity" is the Pythagorean theorem, applied to a triangle inside a unit circle. It is not a new fact. It is an old fact — one you already know from the geometry essay — wearing a costume.4

Before I show you this confirmed, try it in the interactive below. Drag the point to any angle. Watch sine and cosine update. Now — what do you think $\sin^2(\theta) + \cos^2(\theta)$ equals?

The answer, of course, is 1. At 30 degrees: $\sin^2(30°) + \cos^2(30°) = 0.25 + 0.75 = 1$. At 45 degrees: $0.5 + 0.5 = 1$. At any angle: always 1. Because the point is always on the circle, always at distance 1 from the center.

But this identity is not just a verification exercise. It is a constraint. If I tell you that $\sin\theta = 0.6$, you can immediately compute $\cos\theta = \pm\sqrt{1 - 0.36} = \pm 0.8$. The two ratios are not independent — the circle ties them together. Knowing one determines the other, up to a sign that tells you which side of the circle you are on.

This is what I mean by "moral" understanding versus memorization. You do not need to memorize that $\sin^2 + \cos^2 = 1$. You need to see the triangle inside the circle and remember that $a^2 + b^2 = c^2$. The identity writes itself.

The Pythagorean identity tells you that sine and cosine are permanently linked — knowing one determines the other. They are two coordinates of a point constrained to a circle, and the circle holds them in a permanent relationship.

But here is the question that matters most: why does this same wave shape appear everywhere in the physical world?

The Shape of Things That Repeat #

You have seen that a wave comes from a circle. Here is why that matters: the physical world is full of circles.

Not literal circles, necessarily. But processes that cycle — that go around and come back, over and over. A weight bouncing on a spring. A pendulum swinging. The vibration of a guitar string. The alternating current in the power line feeding your screen right now.

Take the spring. Attach a weight. Pull it down and let go. It bounces up, overshoots, comes back down, overshoots again. If you track the height of that weight over time, the curve it traces is a sine wave.

Why? Think about the orbiting point on the circle and imagine looking at it from the side — seeing only its height. It rises, slows down, stops at the top, reverses, speeds up through the middle, slows again at the bottom. That pattern of acceleration — fastest in the middle, momentarily stopped at the extremes — is exactly what a spring does.

The spring pulls hardest when it is stretched farthest, and that pattern of force — always proportional to displacement, always pointing back toward the center — produces exactly the same pattern of motion as the orbiting point on a circle. The farther you pull, the harder it snaps back. That is the signature of the sine wave: the restoring force is proportional to how far you have strayed, and the resulting motion traces the same height-versus-time curve as a point going around a circle.

The spring is not literally going in a circle. But the math does not care. The pattern of speeding up and slowing down is identical, and the trace over time is the same wave.

Here is another example. A pure musical tone — the kind a tuning fork produces — is a sine wave of air pressure. When you hear the note known as concert A (440 Hz), the air pressure at your eardrum is oscillating 440 times per second — a point making 440 trips around the circle every second. The shape of that oscillation is precisely the sine wave you traced on the unit circle.

The wave shape you watched emerge from a circle IS the shape of sound.

This is not a metaphor. When we say "sound is a sine wave," we mean that the mathematical function describing the pressure variations is the same function — $\sin(\theta)$ — that gives the height of a point on a circle. The circle is not an analogy for the wave. It is the mechanism that produces it.

And here is the thought that opens the door to everything that comes after.

So far, we have been looking at one angle, one triangle, one wave at a time. Every time you want to rotate something, you need a sine and a cosine. Rotate a point — one sine, one cosine. Rotate a line segment — you need the same operation for both endpoints. Rotate an entire image — thousands of points, each one needing the same pair of numbers applied in the same way.

Doing this one point at a time is exhausting. What if there were a single object — a compact machine — that encoded the entire rotation and could be applied to any point at once?

That machine is called a matrix. And the entries of a rotation matrix? Cosine and sine.

That is the subject of the next essay.

The Closing Reframe #

All of it — the ratios, the circle, the wave, the sound in your ears right now — flowed from one observation: a comparison between two sides of a triangle does not change when you resize the triangle. It is just comparing. And the comparisons, it turns out, describe the world.

We have covered sine, cosine, and tangent for right triangles, extended them to all angles via the unit circle, and seen why waves emerge from circular motion. There is much more — identities that relate these ratios in dozens of ways, extensions to non-right triangles, and an entire theory (Fourier analysis) that decomposes any periodic signal into sine waves. Those are stories for another time.

What matters here is the foundation: a comparison between two sides of a triangle, invariant under scaling, that connects geometry to the physical world.

But one door remains open. You have one comparison — a ratio in a single triangle. In the next essay, you will have a machine that applies many comparisons at once, transforming entire coordinate systems in a single operation. That machine is a matrix. And the entries of a rotation matrix — the numbers that tell it how far to turn — are the cosine and sine you just spent this whole essay getting to know.