a mysterious complex formula

a mysterious complex equation. Solving cos sin = i by using complex exponentials and the quadratic formula. Euler's identity and imaginary numbers will come into play here as well as the double angle formula sin(2x) = 2 sin(x) cos(x) The solutions will lie on two parallel lines which is really cool. This is a must see for any calculus and complex analysis student! Enjoy this cos(x) sin(x) = i adventure
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[@iabervon]

Even in complex numbers, sin(2z) is periodic with period pi, so you pretty much have to get a picture like that. People generally know about sin(2z)=2w when w happens to be a real number between -1/2 and 1/2, and the two sets of dots just go off of the real line when you leave that domain, while still being two sets of dots each with that spacing.

[@guillermo3412]

Very cool stuff! 👍👍😀

[@TheMemesofDestruction]

How very Fascinating! Thank you Dr. Peyam! ^.^

[@brendanlawlor2214]

nice as usual enjoying yer being back posting regularly ❤8

[@prognosticouture1011]

Very cool! I didn't see the simple double angle formula and decided to substitute sin(z) for √(1-cos^2(z)). I'm still working on the solution, but it appears that the golden ratio is involved, along with the complex arccos function

[@michaelbaum6796]

Very nice equation, thanks a lot👍

[@alexdemoura9972]

"multiples of myself" 😁😆😄
So, from Dr. Peyam's point of view - there are four: 2 pi M. i = two (Peyam an' I) = 4 Peyams.
Put all of them to work, Dr. Peyam - three more videos, please.

[@alipourzand6499]

I sure do want more math like that! The only place to see this kind of math 🔜

[@armanavagyan1876]

PROF thanks a lot)

[@GreenMeansGOF]

Please finish the video by writing out all solutions together so that we can compare them. Thank you. Keep up the great work.

[@drpeyam]

Thank you but I’m confused what you mean

[@GreenMeansGOF]

@@drpeyam For example, at the end, we have the formula for the case 2 solutions but you erased the case 1 solutions. It would be nice if you rewrote the case 1 solutions so that we have all the solutions written on the board at the same time. I hope that clarifies the confusion.

[@General12th]

Nifty! I'm going into math and I should probably memorize all the angle identities so I can solve problems like this better.

[@yoav613]

Yay we have pim😃

[@etiennebasset7493]

Complex is beautiful😊

[@snejpu2508]

Pi am I? No, pi are you. :-)

[@drpeyam]

Hahaha

[@zgr-h8309]

I found 8 more valuse:
Z=-iln(+-1+-sqrt(+-sqrt(3)i)
+-= Plus or minuse

[@drpeyam]

Too many +/-

[@wojtech6221]

I really like your content. 🙂 I understand everything. Do you work at the university? Big greetings from Poland! 🙂📏📐📈📉👍

[@drpeyam]

Thank you!!! I work at Brown University :)

[@aarongleitman1949]

I do not quite understand how2piMi works, but this is very fascinating. Thank you.

[@anarchy5369]

Periodicity of a unit circle where M is just some integer. Every 2pi, you get to the same point you started at

[@justintroyka8855]

What a fun problem! I think you made a small error when drawing the solutions in the complex plane: case 1 is above the real axis, not below. This is because ln(-2+√5) is negative, so -(1/2)*ln(-2+√5) is positive. In fact, -2+√5 = (2+√5)^-1, so -(1/2)*ln(-2+√5) = (1/2)*ln(2+√5). Thus the imaginary part of the case 2 solution is exactly the negative of the imaginary part of the case 2 solution. Isn't that fun?

[@samosamo4019]

Mysterious!!!! Please change the title

[@drpeyam]

Why?

[@samosamo4019]

@@drpeyam qualifying as mysterious!

[@drpeyam]

It is mysterious!