This might be a good thread to ask this. I have done all the math in college, but I don't fundamentally understand sine, cosine, tangents, and logs. For sine, cosine, and tangents, I understand how they may correlate on a graph, but that doesn't really give me additional insight into how to apply or use any of this. Similar thoughts on logs, especially when it comes down to O(logn).
Maybe someone can direct me to a video to help me develop a deeper insight?
Consider following John von Neumann's advice "in mathematics you don't understand things. You just get used to them." Write down the first 100 equations involving sin, cos, tan, log. Use those equations in 100 larger derivations. Visualize the functions in 100 different ways (2D, 3D, animated).
It's a lot of memorization and "getting used to" the math, punctuated by a few "aha moments" of insight when you realize the 100 examples can be compressed into say only 50 examples. In retrospect its tempting to think the insights led to understanding, but really they were just the dopamine hits on top of the real stuff of understanding: tedious acclimation to a new abstract realm.
I don't know exactly from what angle you're looking at this, so let me explain through trigonometry.
We know that triangles may be displaced, rotated, flipped and scaled while still looking the same. We have a word for this: we say two triangles are congruent when they are the same up to these operations.
This means that there is something intrinsically invariant about triangles. Can we find it? Actually yes! If the sides of a triangle have lengths A, B, C, then the ratios A/B, B/C, etc. are invariant. That is, if you make a triangle twice as big, all sides will multiply by 2, so A/B becomes (2A/2B)=A/B -- it's the same!
So, we can come up with names for these invariant ratios. They're most useful for right triangles. To give names, we need to pick any of the two smaller angles as a reference, call it "a". Now, let C be the largest side, A be the side opposite to the angle and B the adjacent side. Then, the ratio A/C can be called sin(a), B/C can be called cos(a) and A/B can be called tan(a).
Since sin(a), cos(a) and tan(a) are ratios, they only depend on the angle, not how big your triangle is. But if you know some side of the triangle, then you can know all of the other using these values. So sin, cos and than really capture the uniqueness I was talking about!
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Now the applications. I memorized these as the "divide by C" rule.
The Pythagorean theorem says thar A^2 + B^2 = C^2.
Divide by C^2, and you get sin(a)^2 + cos(a)^2 = 1.
Divide by cos(a)^2, and you get tan(a)^2 + 1 = sec(a)^2.
These are all the Pythagorean theorem on disguise.
Your "divide by C" rule is awesome, totally stealing this! I also never really grokked trigonometry, but your invariant explanation made it click instantly, wow.
That's kind of what I mean. It's just a mechanical equation and that's how I understand it. They are all ratios, but is that all it is? what insight can I get from that and why do I care about its application in the workforce? when would I ever need such functions? etc...
Maybe someone can direct me to a video to help me develop a deeper insight?