Truly Understand Trigonometry
What this covers
Starting with the Babylonians, we rediscover Angles, Triangles, Circles & Ratios to derive the beautiful Math of a Wave.
❤️ *Credits* Babylon pic: https://medium.com/@Mohammedthanvir/hammurabi-1810-1750-bce-babylonian-law-pioneer-3347b9ca7e72 Numerals pic: https://de.wikipedia.org/wiki/Datei:Babylonian_numerals.svg Song: https://www.youtube.com/watch?v=5py6E6yo7wk (CC BY-SA 4.0) Song: Out To The World - Axon Terminal Song: https://www.youtube.com/watch?v=b0q5PR1xpA0 (CC BY-SA 4.0)
🎬 *Making-of* I code the Animations in Python, especially by using this amazing library: https://www.manim.community/ (initially developed by https://youtube.com/c/3blue1brown) You can find my code here: https://github.com/snsus/youtube https://gitlab.com/snsus-code/youtube
🫂 *Social* Mastodon: https://mastodon.social/@snsus Bluesky: https://bsky.app/profile/snsus.bsky.social Instagram: https://instagram.com/snsus.qed
☕ *You would support me?* https://ko-fi.com/snsus
00:00 Intro 00:31 What is an angle? 03:39 The right-angled Triangle 06:34 Waves
#science #math #education #trigonometry #triangle #wave #sin #cosine #tan #angle #stem #babylon #circle #pi #ratio
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Trigonometric functions (sine and cosine) arise naturally from the geometry of right triangles and the unit circle, and these functions describe wave behavior mathematically by encoding how side lengths change as angles vary.
- Trigonometric ratios (sine, cosine, etc.) are invariant properties of triangles that depend only on angle, not size
- When plotted against angle, sine and cosine produce wave-like curves that can be manipulated to model real-world waves
- The unit circle provides a geometric template that unifies angle measurement with arc length, enabling the visualization of trigonometric functions as functions of angle
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In nature, numerous phenomena propagate in the form of waves or can be described by the concept of a wave, and these phenomena can be modeled mathematically by tracing their description back to a simple geometric shape: the triangle.
“In nature, we observe numerous phenomena that propagate in the form of waves or can be described by the concept of a wave. To understand these phenomena, we model them mathematically. It is remarkable that the mathematical description of a wave even when solving complicated differential equations can be traced back to a simple geometric shape, the triangle.”
The unit circle (a circle with radius 1) was adopted as a template for measuring angles: instead of dividing an angle into discrete parts (like the Babylonian 60 parts), the arc length enclosed by the angle on the unit circle is measured, so a 60° angle corresponds to 1/6 of the circumference, which equals π/3 radians.
“Instead of imagining that, for example, a 60° angle consists of 60 parts, we place our circle template with its center at the tip of the angle and measure the length of the ark that this angle encloses. In the case of this angle, the ark length corresponds to 16th of the entire circumference. That is 2 pi / 6 or p [π]/3 in reduced form.”
The right triangle became the preferred triangle in mathematics (replacing the equilateral triangle) because it can represent all other triangles (any triangle can be constructed from two right triangles), the Pythagorean theorem applies to it, and understanding right triangles enables understanding all triangle types.
“Oh, and the coolest triangle was no longer the equilateral triangle, but the right angle triangle because it can do everything that other triangles can do. And in addition, the Pythagorean theorem applies to it. Furthermore, we can construct any other triangle from two right angled ones. So if we understand the right angle triangle, we understand all the others.”
Trigonometric ratios (sine, cosine, etc.) are invariant: they produce the same numerical value regardless of the size of the triangle, because changing the ratio requires changing only the numerator or only the denominator of the ratio, which is impossible without also changing the other, unless the angle itself changes.
“And now comes the remarkable part. As with the circle, these ratios also correspond to a number that does not change no matter how large or small we make the triangle. For the number to change, for example, in the first ratio, we would have to change only the numerator or only the denominator. But how can we change the numerator, which is the length of the opposite side, without changing the denominator, which is the length of the hypotenuse? Well, we would have to change the angle alpha.”
Because trigonometric ratios change only when the angle alpha changes, they can be understood as functions that depend on alpha; each ratio has been assigned its own function name: sine, cosine, tangent, cotangent, secant, and cosecant.
“So, if these ratios only change when alpha changes, then we can also understand them as functions that depend on alpha. Each of these functions has been given its own name. The first one is called s the next one cosine. Then there's tangent, co-angent, seccant and cosecant.”
If the hypotenuse of a right triangle is set to a unit length (length 1), then sine(alpha) simply equals the length of the opposite side, cosine(alpha) equals the length of the adjacent side, and all other trigonometric ratios can be calculated using only sine and cosine.
“If we choose a fixed length for the hypotenuse, then we might as well choose a length that simplifies everything a little bit. For instance, with a length of one, the sign of alpha simply corresponds to the length of the opposite side and the cosine to the length of the adjacent side. And even better we see that we can calculate the other ratios using just the sign and cosine.”
The unit circle template can be used to measure angle alpha: with a unit hypotenuse and right triangle inscribed in the unit circle, sine(alpha) represents the y-coordinate (opposite side) and cosine(alpha) represents the x-coordinate (adjacent side) of the point on the unit circle.
“And we can now also use our unit circle template to measure the angle alpha.”
To visualize trigonometric functions, a coordinate system can be constructed with angle (alpha) on the x-axis and the trigonometric ratio value on the y-axis; as the angle increases from 0 toward 90°, the sine value increases from 0 toward 1.
“To do this, let's draw a coordinate system and plot the angle on the x-axis and the length of the opposite side on the y-axis. If we increase the angle slightly, the opposite side becomes longer. As we approach an angle of zero, the length becomes smaller and smaller. At an angle of exactly zero, the opposite side disappears. But we can imagine that it's still there just with a length of zero.”
When tracing the sine function around a complete rotation of the unit circle, the resulting curve resembles a wave; negative values in this representation indicate that the opposite side is in the lower semicircle of the circle.
“If we now rotate around the entire circle and trace the path, we obtain this curve for the sign. Here, negative lengths mean that the opposite side is in the lower semicircle.”
The cosine function can be graphed in the same coordinate system as sine, and it has the same wave shape as sine but is offset (phase-shifted) by 90° (or π/2 radians), because the adjacent side is geometrically offset by 90° from the opposite side.
“For the cosine, we can simply copy the coordinate system and offset it by 90° because the adjacent side is offset by 90° to the opposite side. However, we can also draw the cosine function in the same coordinate system. Here we can also see that the cosine looks exactly like the sign. It's just shifted by 90° or half of pi.”
If we understand sine, we also understand cosine because cosine is simply sine shifted by 90°.
“So if we understand the sign, we also understand the cosine.”
The sine function can be extended beyond a single rotation of the unit circle (0 to 2π radians) by continuing to rotate beyond 2π, with the wave pattern repeating periodically.
“We can just agree that we can keep rotating around the circle beyond 2 pi. And a negative angle just means we rotate clockwise.”
Not all natural waves are pure sine waves; however, the sine function can be manipulated with additional parameters to model real-world waves by modifying the amplitude (height of the wave crest), frequency (how quickly the wave oscillates), and phase (horizontal or vertical shift of the function).
“In reality, of course, not all waves look like this pure sine wave. But we can manipulate this function with additional numbers. For example, if we multiply the sign by a certain number, we influence the height of a wave crest, the so-called amplitude. And if we multiply the angle by a certain number, we influence its frequency. We can also shift the function in the x or y direction or even put alpha into a separate function.”
The Babylonians used a sexagesimal (base-60) number system and applied this to dividing space and time: they divided angles into 60 parts (degrees) and because exactly six angles of 60° form a complete circle, a full circle measures 360°.
“because these people used the sexesimal system for the concept of numbers, which is a number system based on the number 60, they loved to divide different things into 60 parts. Not only were units of time divided into 60 parts, but also units of space. So they imagined that this angle also consisted of 60 equal parts. One angle therefore corresponds to 60 um angle parts which we now call degrees and symbolize with a circle. Since exactly six of these form a complete circle, we measure 6 * 60 or 360° in a circle.”
When humans later adopted the decimal (base-10) number system based on having 10 fingers, they were dissatisfied with the Babylonian sexagesimal division of space and time, so they changed angle measurement using pi, the ratio of a circle's circumference to its diameter, which is constant regardless of circle size.
“When humans later discovered that they had 10 fingers and began using the decimal number system, they were not very amused by the fact that the Babylonians divided spacetime by 60. So they divided the units of time further into powers of 10 and the angle measurement also had to be changed. For this purpose, a fascinating property of a circle was used. If we divide the length of the circumference C by the length of the diameter D, that is if we calculate the ratio of circumference to diameter, we get a number that always remains the same no matter how large or small we make the circle. Today, we know this number as pi.”
A full circle measures 2π radians, corresponding to the complete circumference of the unit circle.
“And for a full circle, we eventually measured the complete circumference of 2 pi.”
The Babylonians formulated the concept of an angle based on the equilateral triangle, observing the space between two sides as defining one angle.
“for the Babylonians, it was the equilateral triangle. They were so impressed by it that they even formulated the concept of an angle based on it. In a sense, what they observed between two sides was one angle.”
Pi is a constant ratio of a circle's circumference to its diameter (C/D) that remains the same regardless of circle size; when rearranged with diameter = 2 × radius, the circumference formula becomes C = 2π × radius, so for a unit circle with radius 1, the circumference is 2π.
“If we rearrange the equation and set the diameter equal to 2 * the radius, the circumference corresponds to 2 pi * the radius. with a radius of 1. The circumference is simply 2 * pi.”
A 90° angle corresponds to 1/4 of the unit circle's circumference, which equals π/2 radians; this angle is so frequently used in mathematics that it is marked with a distinctive dot symbol.
“At a 90° angle, the ark occupies a quarter of the circle. So this angle corresponds to a quarter of 2 pi or half of pi for short. Because we use this angle so often and want to recognize it quickly, we mark it with an additional dot.”
In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called legs; from the perspective of a specific acute angle (alpha), the legs can be further named as the opposite leg (across from alpha) and the adjacent leg (next to alpha).
“All right, let's first name the individual sides. The side opposite the right angle is called the hypotenuse. The other ones are called legs. From the perspective of an angle, which I will simply call alpha, we can also name the legs more precisely. The side opposite to alpha is then called the opposite leg. and the one adjacent to the angle is called the adjacent leg.”
Six trigonometric ratios can be formed from the three sides of a right triangle (sine, cosine, tangent, cotangent, secant, and cosecant), where the last three are reciprocals of the first three (with swapped numerator and denominator).
“With these three sides, we can create a total of six ratios whereby in the last three ratios only the numerator and the denominator of the first three are swapped.”
A negative angle in unit circle trigonometry means rotating clockwise instead of counterclockwise.
“And a negative angle just means we rotate clockwise.”
And that looks like a wave, doesn't it?
“And that looks like a wave, doesn't it?”