Math for fun, sin(sin(z))=1

The trigonometric equation sin(z)=1 is fairly easy to solve but not sin(sin(z))=1. Here we will be using the complex exponential definition of sine, which is from Euler's formula e^(i*theta)=cos(theta)+i*sin(theta), to solve this equation. We will see sin(sinz)=1 actually has infinitely many complex solutions, just like how we solve sin(z)=2.

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Thank you,
blackpenredpen

[tibees]

I was so focussed on the board I didn't see the shirt until I saw a comment 😂Thanks for wearing it!

[mrmimeisfunny]

India: We use colors as variables.
Arabia: Well, we don't really want to mess with different pigments while doing math. We're just going to use letters.
Europe: The letters are not Christian enough, we will use our own latin and greek letters.
Blackpenredpen: 🙂

[dorian]

For happy face^2 it should've been drawn as an actual square face, for the true immersion.

[agabe_]

You know its terrifying when he giggles time to time.

[BlissOn]

Algebra: letters as variables
Trigonometric algebra: latin letters as variables
Blackpenredpen algebra: emojis

[jon]

2:52 "I'm just going to put down a happy face....because a fish is too difficult."
🤔 but a fish can be represented with just one letter: 𝛼

[ericbright]

"This looks... yeah." Sums it up quite nicely.

[TheMartian]

My guy here rockin' that Tibees merch while solving these unholy equations

[breadlegend]

You know its worse when he has more than 2 pens

[caseyleung]

7:05 Actually, ln(x+sqrt(x^2-1)) = arccosh(x), which can further simply the answer.

[Vampianist]

6:15
and that’s why most people don’t have happy faces when they do maths

[SHASHANKRUSTAGII]

your beard looks like a perfect binary tree

[christopherdyson]

This is completely unrelated, but I was trying to figure out transistors earlier today since one of the bonus problems in my principles of electrical engineering textbook had them in an example of a monostable vibrator (I havent exactly seen a transistor before in problems... or real life... not even sure why it brought them up because the problems were about basics of DC RC circuits)

But apparently the way to calculate the voltage across transistors uses the Lambert-W function and I thought back to your "fish" videos you did on the Lambert-W function. Honestly I didn't know it had much use out of "math for fun".

[matthewstevens]

The video on sin(z)=2 was the first video I saw from your channel! I've been following you since that video was uploaded ;)

[imaginaryangle]

I love how you use basic examples to train all the little concerns and caveats that must be observed when solving a specific class of problems. Very effective and fun to watch, especially with emoji substitution 😄

[Eichro]

"We'll have to go to the complex world"
*[screams in agony]*

[Gamiboi]

I’m surprised how I’m slowly starting to understand these types of videos as I learn. I still remember how I would not understand any statements in these videos a few years ago.

[nombreusering]

I remember u used this technique/Other way to write arcsin in ur sin(?)=2 vid. Amazing

[neilgerace]

Happy face, meet fish. Fish, meet happy face.

[AnCoSt]

I think it'd be cool if you quickly calculated the solution for n, m=0, to show what value for z that would present. I absolutely love these complicated problems that you keep showing!! Any plans to consider more AIME or even USAMO/IMO videos?