30 Nov 2021 |

milia | oh boy | 01:01:04 |

milia | from one awkward moment to the next | 01:01:11 |

ISO 4683-1#3216 | uh-oh | 01:01:11 |

ISO 4683-1#3216 | lol | 01:01:14 |

milia | sry 😐 | 01:01:14 |

ISO 4683-1#3216 | you're fine | 01:01:18 |

milia | 🙂 | 01:01:23 |

milia | so, this is due to E. Landau | 01:01:29 |

milia | 1 = cos(0) = cos(x-x) = cos(x)*cos(x) + sin(x)*sin(x) | 01:01:51 |

Garklein#9297 | you can escape *s with \ | 01:02:05 |

milia | * 1 = cos(0) = cos(x-x) = cos(x)*cos(x) + sin(x)*sin(x) | 01:02:20 |

milia | tx 🙂 | 01:02:24 |

ISO 4683-1#3216 | neat | 01:02:57 |

milia | 🙂 | 01:03:03 |

Garklein#9297 | wait how do you get that last part | 01:03:15 |

milia | you'll find it in his integral + differential calculus book | 01:03:21 |

ISO 4683-1#3216 | cos(a-b) = cos(a)*cos(b) + sin(b)*sin(a) | 01:03:40 |

milia | it's the trig identity cos(a-b) = cosa * cosb + sina * sinb | 01:03:42 |

milia | yeh | 01:03:47 |

Garklein#9297 | oh | 01:03:55 |

milia | what blew my mind was the cos(0) = cos(x-x) part | 01:04:19 |

milia | pretty creative moment | 01:04:26 |

milia | creating a solution out of zero | 01:04:33 |

ISO 4683-1#3216 | mhm | 01:04:51 |

Marshall | Follows pretty naturally from the complex unit circle: conjugate of exp(ix) is exp(-ix), so magnitude-squared of exp(ix) is exp(ix)exp(-ix) or exp(i(x-x)). | 01:10:31 |

Marshall | Huh, the typical way you'd prove exp(ix) has magnitude 1 for real x is to take the derivative of exp(ix)exp(-ix), knowing that exp(z) is its own derivative by definition (you also need exp(0)=1). Same thing works for cos(x)cos(x) + sin(x)sin(x), since the cos term gives you the negative of the sin one. | 01:15:12 |

milia | oh yes, I assume you've used the euler formula at some point ? | 01:52:11 |

Marshall | I take that to be the definition of sine and cosine, so yes. | 02:07:45 |

milia | Nice. I think in the case of Landau he took the taylor expansions as the definitions of the sine and cosine functions. | 02:37:13 |

Marshall | Makes sense. I think proving existence of a solution to the exp(z) differential equation might have been complicated to do rigorously at the time? Then again I don't remember how to prove that at all. | 02:54:08 |