Trigonometry

the fuckin triangle yo

The Simplest Shape

Why triangles?

Seriously, of all the shapes, why does an entire branch of math obsess over three-sided polygons?

Because triangles are rigid.

Take four sticks and pin them together at the corners. You get a square, but it's floppy—push on one corner and it collapses into a rhombus. The angles can change without the sticks breaking.

Now take three sticks and pin them together. Try to push on it. You can't. The triangle holds its shape. The angles are locked the moment you fix the side lengths.

This is why bridges are made of triangles. Why roof trusses are triangles. Why structural engineers think in triangles. It's the minimal rigid structure.

And here's the thing: any polygon can be broken into triangles. A square is two triangles. A pentagon is three. Any shape, no matter how complex, can be triangulated. Solve the triangles, you've solved the shape.

Triangles are the atoms of geometry.

Pythagoras: The Original Math Cult Leader

Around 500 BCE, a Greek philosopher named Pythagoras started a school. But "school" doesn't capture it. It was more like a religious commune. The Pythagoreans had secret rituals, dietary restrictions (famously, no beans), and a core belief that would shape mathematics forever:

All is number.

They believed numbers weren't just useful for counting—numbers were the fundamental substance of reality. The universe wasn't made of earth, water, air, and fire. It was made of mathematical relationships.

Sounds mystical. Sounds like woo. But here's the thing: they weren't entirely wrong.

The Pythagoreans discovered that musical harmony is mathematical. A string half as long vibrates at twice the frequency—that's an octave. A string 2/3 as long gives you a perfect fifth. The notes that sound good together are the ones with simple numerical ratios. Music is math made audible.

This blew their minds. If something as seemingly subjective as "sounds good" could be reduced to ratios of whole numbers, what else was secretly mathematical?

The Theorem

The Pythagorean theorem probably wasn't discovered by Pythagoras. The Babylonians knew it a thousand years earlier. But his school proved it rigorously and attached his name to it forever.

You know it: a² + b² = c²

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

This isn't just a formula. It's the definition of distance. It's why the coordinate plane works. It's the foundation of trigonometry. It generalizes to any number of dimensions. It's arguably the most important equation in geometry.

The Pythagoreans loved it because it connected geometry to arithmetic. Shapes and numbers, unified. All is number, proven.

The Crisis

Then one of them—tradition says it was Hippasus—discovered something horrifying.

The diagonal of a square with side 1 has length √2. And √2 is not a ratio of whole numbers. It's irrational. It cannot be expressed as a fraction. Its decimal expansion goes on forever without repeating.

This broke the Pythagorean worldview. If all is number, and numbers mean whole numbers and their ratios, then what the hell is √2? It exists—you can draw it, measure it, see it—but it's not a "number" by their definition.

Legend says the Pythagoreans drowned Hippasus for revealing this scandal. Probably not true, but the story captures how existentially threatening the discovery was. The foundation had a crack.

Math survived. The concept of number expanded to include irrationals. But that tension—between the discrete and the continuous, between whole numbers and the infinite decimals between them—never fully went away. It echoes through calculus, through real analysis, through set theory. The Pythagoreans found the crack early.

The Legacy

Pythagoras matters because he was right about the important thing: mathematical relationships really are woven into reality. Not just as descriptions, but as structure.

The orbits of planets follow mathematical laws. Atoms bond in geometric patterns. Quantum mechanics runs on linear algebra. Evolution follows game theory. Everywhere we look, we find math.

"All is number" was too simple. But "all is structure" or "all is relationship"—that holds up. The Pythagoreans saw it first.

And the theorem that bears his name? It's about to become the backbone of everything in this chapter. Every trig function, every identity, every application—it all traces back to a² + b² = c².

Angles and Ratios

Here's what trigonometry actually is: the relationship between angles and side lengths.

You got a right triangle—one corner is 90 degrees. The longest side (across from the right angle) is the hypotenuse. The other two sides are called opposite and adjacent, relative to whatever angle you're looking at.

The core insight: if you know the angles, you know the RATIOS of the sides. Not the actual lengths—you need at least one length for that—but how they compare to each other.

Fix an angle. Any right triangle with that angle, no matter how big or small, will have the same ratio of opposite to hypotenuse. The same ratio of adjacent to hypotenuse. The same ratio of opposite to adjacent.

Those ratios get names:

Sine (sin) = opposite / hypotenuse

Cosine (cos) = adjacent / hypotenuse

Tangent (tan) = opposite / adjacent

That's it. That's the core of trig. Everything else builds from these three ratios.

The Unit Circle

Now here's where it gets beautiful.

Draw a circle with radius 1. Put the center at the origin of a coordinate plane. This is the unit circle.

Pick any point on the circle. Draw a line from the center to that point. That line makes an angle with the positive x-axis—call it θ (theta).

Now drop a vertical line from your point down to the x-axis. You just made a right triangle. The hypotenuse is 1 (the radius). The horizontal leg is how far over the point is—that's the x-coordinate. The vertical leg is how far up the point is—that's the y-coordinate.

What's the cosine of θ? Adjacent over hypotenuse. That's x/1 = x.

What's the sine of θ? Opposite over hypotenuse. That's y/1 = y.

The point's coordinates ARE (cos θ, sin θ).

Let that sink in. Every point on the unit circle is just (cosine, sine) for some angle. Sine and cosine aren't abstract functions—they're literally "where am I on the circle?" Cosine is how far left or right. Sine is how far up or down.

I saw the sine, and it opened up my eyes. I saw the sine.

Ace of Base was onto something. Once you see that sine is just vertical position on a circle, everything changes.

Waves Are Circles

Now spin around the circle at constant speed. Your angle θ increases steadily with time.

Your x-coordinate—cos(θ)—oscillates back and forth. Right, center, left, center, right, center, left...

Your y-coordinate—sin(θ)—oscillates up and down. Up, center, down, center, up, center, down...

Plot these against time and you get waves. Smooth, undulating, repeating curves. The sine wave. The cosine wave.

Every wave is a circle unrolled over time.

Sound is pressure waves—oscillating air molecules. Light is electromagnetic waves—oscillating electric and magnetic fields. Ocean waves, radio waves, AC electrical current, the vibration of a guitar string—all of them are circles in disguise.

When you hear a pure tone, you're hearing a sine wave. When you see a color, you're seeing an electromagnetic sine wave at a particular frequency. Circles everywhere, hidden in plain sight.

Why Radians?

You probably learned angles in degrees. 360° in a full circle. That works, but it's arbitrary—why 360? (Blame the Babylonians and their base-60 obsession.)

Radians are better. Here's why:

One radian is the angle where the arc length equals the radius. Wrap the radius around the circumference—the angle you've swept out is one radian.

A full circle has circumference 2πr. If the radius is 1, the circumference is 2π. So a full turn is 2π radians. Half turn is π radians. Quarter turn is π/2 radians.

Why is this better? Because it makes the math clean. The derivative of sin(x) is cos(x)—but only if x is in radians. In degrees, you get ugly conversion factors everywhere.

Radians are the natural unit for angles because they tie angle directly to arc length. No arbitrary constants. Just circles being circles.

The Pythagorean Identity

Remember the unit circle point: (cos θ, sin θ).

The distance from the origin to that point is 1 (it's on the unit circle). By the Pythagorean theorem:

cos²θ + sin²θ = 1

Always. For any angle. This is the Pythagorean identity, and it's the most important equation in trig.

It says sine and cosine are bound together. If you know one, you can find the other (up to sign). They're not independent—they're two coordinates of the same point on a circle.

This connects back to Noether and Yin Yang. Sine and cosine aren't opposites—they're complements. They're the same circular motion viewed from perpendicular directions. Shift your perspective 90 degrees and sine becomes cosine, cosine becomes sine.

Triangles to Circles to Waves

So here's the progression:

Start with a triangle. The simplest rigid shape. Angles determine ratios.

Put the triangle in a circle. Now the ratios become coordinates. Sine and cosine are positions on the circle.

Spin the circle. Now position becomes oscillation. Sine and cosine are waves.

Triangle → Circle → Wave.

Static → Rotating → Oscillating.

Same structure, different views. Trig is the bridge between shapes and motion.

The Other Trig Functions

Sine, cosine, tangent are the main three. But there are three more—the reciprocals:

Cosecant (csc) = 1/sin = hypotenuse / opposite

Secant (sec) = 1/cos = hypotenuse / adjacent

Cotangent (cot) = 1/tan = adjacent / opposite

These show up less often but they're useful in certain situations. They're not new information—just different ways to express the same ratios.

Tangent: The Slope

Tangent is special. It's sin/cos, which is y/x on the unit circle. That's the slope of the line from the origin to your point.

As you go around the circle:

At 0°, tangent is 0 (horizontal line)

At 45°, tangent is 1 (diagonal line, slope 1)

At 90°, tangent is undefined (vertical line, infinite slope)

Tangent blows up at 90° because you're dividing by cos(90°) = 0. The line becomes vertical. This is where the tangent function has its asymptotes—the places it shoots off to infinity.

The name "tangent" comes from the tangent line to the circle. There's a geometric construction where the tangent function literally measures the length of a tangent line segment. The word means "touching"—the tangent line touches the circle at exactly one point.

Inverse Trig: Going Backwards

Sometimes you know the ratio and want the angle.

If sin(θ) = 0.5, what's θ?

The inverse sine function—written sin⁻¹ or arcsin—gives you the answer: θ = 30° or π/6 radians.

But there's a catch. Sine of 30° is 0.5. Sine of 150° is also 0.5. So is sine of 390°, and 510°, and infinitely many other angles.

The inverse functions have to pick one answer, so they're restricted to specific ranges. Arcsin gives you an angle between -90° and 90°. Arccos gives you between 0° and 180°. Arctan gives you between -90° and 90°.

This is a recurring theme in math: when you invert a function, you often have to deal with multiple possible answers. The function goes from many inputs to one output; the inverse has to somehow pick which input you meant.

Law of Sines, Law of Cosines

What about triangles that aren't right triangles?

Law of Sines: In any triangle, the ratio of a side to the sine of its opposite angle is constant.

a/sin(A) = b/sin(B) = c/sin(C)

The bigger the angle, the bigger the opposite side. Proportionally.

Law of Cosines: A generalization of the Pythagorean theorem for any triangle.

c² = a² + b² - 2ab·cos(C)

When C = 90°, cos(C) = 0, and this reduces to c² = a² + b². The Pythagorean theorem is a special case.

These two laws let you solve any triangle if you know enough pieces. Three sides? Use law of cosines to find an angle. Two angles and a side? Use law of sines. The triangle can't hide from you.

Fourier: Everything is Waves

Here's maybe the wildest application.

Early 1800s. French mathematician Joseph Fourier is studying heat flow. He discovers something that sounds impossible: any periodic function can be written as a sum of sines and cosines.

Any wave shape—square waves, sawtooth waves, whatever weird oscillation you want—can be built by adding up simple sine waves of different frequencies.

This is the Fourier series. It says that sines and cosines are a basis for periodic functions. Just like any point in 3D space can be written as a combination of x, y, and z directions, any repeating pattern can be written as a combination of sine waves.

Your voice is a complex wave. But it can be decomposed into pure tones—sine waves at different frequencies and amplitudes. That's how audio compression works. That's how equalizers work. That's how noise-canceling headphones work.

Fourier analysis is everywhere: image processing, signal processing, quantum mechanics, data compression. All because sine waves are the atoms of oscillation.

Why Triangles Lead to Everything

Triangles seem simple. Three sides, three angles, rigid structure.

But inside triangles you find circles (the unit circle definition of sine and cosine). Inside circles you find waves (circular motion projected over time). And waves describe everything that oscillates, vibrates, cycles, or repeats.

Sound. Light. Heat. Electricity. The quantum behavior of particles. The orbits of planets. The beating of your heart.

Pythagoras said all is number. He was close. All is relationship. All is ratio. All is structure.

The triangle was a door. Trigonometry is what's on the other side.