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Why sin and cos Shouldn't Be Defined With Triangles

Audience:

Most definitions of sin and cos start with a triangle. That definition breaks completely past 90°.

This video builds sin and cos from scratch using $e^{ix}$ , derives $cos²+sin²=1$ from a single exponent cancellation, and arrives at the Pythagorean theorem as a consequence of scaling and not the other way around.

This is not the simplest proof of Pythagoras. It is a derivation of the functions that make Pythagoras work.

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6 Overall score*

66 Rank

7 Votes

5 Comments

Comments

7.1

An absolute mathematical breakthrough in changing how we perceive trigonometric foundations! Reversing the axiom to derive the Pythagorean theorem via $e^{ix}$ is pure genius and a brilliant out-of-the-box conceptual framework.

However, the execution relied heavily on dense formulas and plain graphs without enough dynamic visual aids to help guide the audience. Additionally, enhancing the clarity of the vocal narration would greatly improve engagement for a broader audience. A highly commendable, philosophically deep, and valuable entry overall!

6.1

The topic is quite interesting, the animation is pretty, and I really like the quote you use in the video.

But I think even if Write() animation is really fun, you use too much of them. And even every part of your video is easy to understand, together It’s not clear why there is so many sudden jump of the topic. So, the organization of the video is quite confusing.

You are a bit confused about the order in which things are introduced in highschool and the circular reaosning is not actually circular. Moreover, going the full route over the explansion definition of e and complex numbers is overkill. Especially when the whole essence of complex numbers is “sin and cos are the coordinates of a point on the unit circle”.

2.9

Great graphics. I prefer more description at the start of the video to motivate the learning. Then ending has a thought provoking statement. I think some of the maths could have been explained better. The use of elements of the Taylor series to show how the circle is constructed was hasty and possibly not providing a suitable explanation for a video.

2.9

I feel this video jumps around to a lot of topics without really tying them into the main topic of the video introduced at the start. For example, you introduce that e^x is its own derivative, but you don’t really make it clear why this is useful. You introduce the e^(i pi) + 1 = 0 formula, but although it’s a neat result, it doesn’t really seem all that relevant here. And the Wheeler quote about GR felt a bit like it came out of nowhere, too.

But I think the main thing you’re missing is an explanation of why sin and cos, which you give as the real and imaginary parts of e^(ix), actually correspond to the geometric notion, in terms of ratios of sides in a right-angled triangle. You have most of the setup you need, but you don’t really take the time to explain it. And to get a proof of the Pythagorean theorem, you do need to know that the sin and cos you’re working with actually do correspond to the triangle-based geometric notion of sin and cos.