Sine function and wave questions

Question

I have been doing research for a few weeks, and I was trying to learn more about sine. I learned how it was used in trigonometry for calculating angles. It is the same as the sine function, which is used in engineering, and it is the same as sound waves. I learned how the function is able to be written on a graph as a wave. I heard about the Taylor series. I think it’s an equation used for calculating the sine of an angle, and when it is plotted on a graph, it is the same as the wave. Q1 Is this a coincidence, or are they the same for a reason? Q2 why is sine related with PI? Q3 (This might be an obvious question) How are functions plotted on graphs.

I have looked very far on the internet and couldn’t find an answer to these questions. I would really appreciate someone to help me. (Also, sorry that I jumbled a bunch of questions into one.)

Answer 1

Q1: Yes, the Taylor series definition, and the trigonometric definition of the sine function are equivalent. There are multiple reasons why this is the case and why this makes sense (many of which you probably haven't seen yet: separation of the complex exponential function into $\cos + i \sin$, differential equations of order 1 and 2, complex numbers in general, Stone-Weierstrass theorem). The most important IMO is the Stone-Weierstrass theorem. To put it in a (caricatured) way which you should be able to understand: see how you can approximate the real numbers via a sequence of rational numbers ? Eg $u_0 = 3; u_1 = 3.1; u_2 = 3.14...$ which converges to $\pi$, "at infinity". Well, there's a theorem (Stone-Weierstrass) that says you can approximate any smooth function (those that have a derivative, and whose derivative has a derivative, etc infinitely onwards) via a sequence of polynomials. "At infinity", this is your Taylor series.

Q2: sine is related to $\pi$ because sine encodes information about rotations, like the complex exponential does, and the cosine function. The reason why you see sine wave in physics is because periodic behavior is also often linked to rotations (through cyclicity). And when you're dealing with rotations and circles in general, you're also dealing with $\pi$.

Q3: plotting functions takes a bit more getting used to, and is very important in mathematics. For functions $\Bbb R \to \Bbb R$, you basically combine your input (a single float) and output (a single float) into a number on the plane, as $(input, output)$. Here are a couple of resources to give you what you need to get the intuitions, since you're curious.

https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr

Gaining an intuition for how changes to the inputs of an equation affect the output

You should also check out Khan Academy if you want some "take-me-by-the-hand" practice.

Answer 2

To really understand the sine function, you need to study it systematically, developing the whole body of knowledge behind this concept. Google searching gives you isolated pieces of information and can be confusing, so you should seek a resource to learn it systematically, preferably with a textbook (if you want a recommendation: James Stewart "Precalculus"), or watch some Khan academy videos.

But if you're curious to know some fun things about the sine function, before your teacher even shows you, we can go for some broad descriptions:

the sine function belongs to the area of mathematics called trigonometry, which is the study of the geometric property called similarity. Let's see how this works, of all plane shapes, the simplest are triangles, since all shapes may be subdivided into triangles. Moreover all triangles may be divided into right triangles by drawing altitudes (heights of triangles), so we should develop the geometry of similarity for right triangles, and this is the perspective of trigonometry.

Every right triangle can be scaled (magnified or microscopi-fied) into a right triangle whose hypotenuse is length 1, and the collection of all right triangles with hypotenuse 1 is best visualized on the unit circle (radius = unit = 1)

each point of the unit circle gives rise to a right triangle whose vertices are the origin, the point on the circle, and the base point on the x-axis. So the collection of all right triangles of hypotenuse 1 are determined by points of the unit circle and the set of all right triangles can be scaled to one on the unit circle.

How do you describe a right triangle at the unit circle? you specify the angle $\theta$ of rotation from the positive x-axis to the point at the unit circle. The hypotenuse is length 1, the legs are length x,y. So we give the definition of two functions

$$\sin\theta = y$$ and $$\cos \theta = x$$

Why do I say trigonometry is the study of similarity? Suppose you wish to reason about a triangle similar to the one at the unit circle with angle $\theta$, that has been scaled by a factor of $r$, so the hypotenuse is $1\times r$, so what are the legs? they are the legs at the unit circle also scaled by $r$ hence the leg opposite to angle $\theta$ is $r\sin\theta$ and the leg adjacent to the angle is $r\cos \theta$. If you now rearrange these expressions, you will understand why textbooks tell you $$\sin \theta=\frac{opposite}{hypotenuse}$$ $$\cos \theta=\frac{adjacent}{hypotenuse}$$ .