I literally suck at explaining but I'll try to point out my thoughts.
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Person: Geometry is about shapes, lines, and so on.

Alien: Oh... Can you show me a line?

Person *looking around*: Umm... see that brick, there? A line is one edge of that brick.

Alien: So lines are part of a shape?

Person: like sort of. Most shapes have lines in them. But a line is a basic concept on its own, a beam of light, a route on a map, or even-

Alien: Bricks have lines. Lines come from bricks. Bricks bricks bricks.

Person: *face-palm*
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Let's build our intuition by seeing sine as its own shape, and then understand how it fits into circles.

Circles have sine but seeing sine inside a circle is like getting the eggs back out of the omelette. It's all mixed together, can't. '-'

Sine that "starts at the max" is called cosine, and secretly it's just a version of sine. Time for both sine waves: put vertical as "sine" and horizontal as "sine'". And we have a circle
A horizontal and vertical "spring" combine to give circular motion. Most textbooks draw the circle and try to extract the sine, but I prefer to build up: start with pure horizontal or vertical motion and add in other.

Though Sine wiggles usually in one dimension and cosine in another.

Circles and squares are a combination of basic components (sines and lines). The circle is made from two connected 1-d waves, each moving the horizontal and vertical direction.

But remember, circles aren't the origin of sines anymore than squares are the origin of lines. They are only examples, not the sources.
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"imagine it takes sine 10 seconds from 0 to max". And now it's pi seconds from 0 to max back to 0? What gives? Sin(x) is the default, off the shelf sine wave that indeed takes pi units of time from 0 to max to 0 (or 2*pi for a complete cycle) sin(2x) is a wave that moves twice as fast.

sin(x/2) is a wave that moves twice as slow.

So, we use sin(n*x) to get a sine wave cycling as fast as we need.

Often, as I know, the phrase "sine wave" is referencing the general shape and not a specific speed.

The Height Of A Triangle - Circle

Sine was first found in triangles. You may remember "SOH CAH TOA" as a mnemonic,
SOH: Sine is Opposite / Hypotenuse
CAH: Cosine is Adjacent / Hypotenuse
TOA: Tangent is Opposite / Adjacent

For a right triangle with angle x, sin(x) is the length of the opposite side divided by the hypotenuse. If we make the hypotenuse 1, we can simplify to:

Sine = Opposite
Cosine = Adjacent
And with more cleverness, we can draw our triangles with hypotenuse 1 in a circle with radius 1. A circle containing all possible right triangles (since they can be scaled up using similarity). These direct manipulations are great for construction (the pyramids won't calculate themselves duh). Unfortunately, after thousands of years we start thinking the meaning of sine is the height of a triangle. No no, it's a shape that shows up in circles (and triangles).

The Infinite Series:

I've avoided the elephant in the room: how in blazes do we actually calculate sine!? Is my calculator drawing a circle and measuring it?

Here's the circleless secret of sine:

Sine is acceleration opposite to your current position.

Using our bank account metaphor, imagine a perverse boss who gives you a raise the exact opposite of your current bank account(that sucks tho) If you have $50 in the bank, then your raise next week is $50. Of course, your income might be $75/week, so you'll still be earning some money $75 - $50 for that week), but eventually your balance will decrease as the "raises" overpower your income.

But don't fear. Once your account hits negative (say you're at $50), then your boss gives a legit $50/week raise. Again, your income might be negative, but eventually the raises will overpower it.

This constant pull towards the center keeps the cycle going when you rise up, the "pull" conspires to pull you in again. It also explains why neutral is the max speed for sine. If you are at the max, you begin falling and accumulating more and more "negative raises" as you plummet. As you pass through then neutral point you are feeling all the negative raises possible (once you cross, you'll start getting positive raises and slowing down).

By the way since sine is acceleration opposite to your current position, and a circle is made up of a horizontal and vertical sine. Circular motion can be described as "a constant pull opposite your current position, towards your horizontal and vertical center".

Geeking Out With Calculus
Let's describe sine with calculus. Like e, we can break sine into smaller effects:

Start at 0 and grow at unit speed
At every instant, get pulled back by negative acceleration.

How should we think about this? See how each effect above changes our distance from center:

Our initial kick increases distance linearly: y (distance from center) = x (time taken)

At any moment, we feel a restoring force of -x. We integrate twice to turn negative acceleration into distance:
-x ={-x^3}{3!}

Seeing how acceleration impacts distance is like seeing how a raise hits your bank account. The "raise" must change your income, and your income changes your bank account (two integrals "up the chain")

So after "x" seconds we might guess that sine is "x" (initial impulse) minus x^3/3! (effect of the acceleration):

Something's wrong -sine doesn't nosedive! With e, we saw that interest earns interest and sine is similar. The restoring force changes our distance by -x^3/3!, which creates another restoring force to consider. Let's consider a spring the pull that yanks you down goes too far which shoots you downward and creates another pull to bring you up (which again goes too far). Springs are crazy but nice.

We need to consider every restoring force:

y = x is our initial motion, which creates a restoring force of impact as I said y = -x^3/3!, which creates a restoring force of impact and y = x^5/5!, which creates a restoring force of impact and y = -x^7/7! which creates a restoring force of impact. Exactly like e, sine can be described with an infinite series

sin(x) = x - {x^3}{3!} + {x^5}{5!} - {x^7}{7!} + ... }

I saw this formula a lot, but it only clicked when I saw sine as a combination of an initial impulse and restoring forces. XDD. The initial push (y = x, going positive) is eventually overcome by a restoring force (which pulls us negative), which is overpowered by its own restoring force (which pulls us positive), and so on.

A few Fun Notes:

Consider the restoring force like positive or negative interest. This makes the sine/e connection in Euler's formula easier to understand. Sine is like e, except sometimes it earns negative interest. There's more to learn here.

For very small angles, y = x is a good guess for sine. We just take the initial impulse and ignore any restoring forces. Summing it up the goal is to move sine from some mathematical trivia (part of a circle) to its own shape,

Sine is a smooth, swaying motion between min (-1) and max (1).

Mathematically, you are accelerating opposite your position. This negative interest keeps sine rocking forever.
Sine happens to appear in circles and triangles (and springs, pendulums, vibrations, sound...etc)

Pi is the time from neutral to neutral in sin(x). Similarly, pi doesn't belong to circles, it just happens to show up there. I don't know how...

Let sine enter your mental notebook (I need a formula to make smooth changes...). Eventually, we will understand the foundations intuitively (e, pi, radians, imaginaries, sine...)  one day and they can be mixed into a scrumptious math salad. :3 fun right?