Fleeting/Random Thoughts

TheEcologist

Global Moderator
Soundtracks that help my writing productivity:

• Star wars I - VI
• Lord of the rings: Fellowship of the ring, Two Towers, return of the king.
• Batman Begins, The Dark Knight, The Dark Knight Rises

Soundtracks that don't help my writing productivity:

• Les Miserables

vinux

Dark Knight
Soundtracks that help my writing productivity:

• Star wars I - VI
• Lord of the rings: Fellowship of the ring, Two Towers, return of the king.
• Batman Begins, The Dark Knight, The Dark Knight Rises

Soundtracks that don't help my writing productivity:

• Les Miserables
I like your selection. Hans Zimmer's soundtracks are good. Whenever I am sad, I play batman's and gladiator's soundtrack.
My favourite is Yanni. This is my default music played during my studies and my experiments.

trinker

ggplot2orBust
Don't forget about the Last of the Mohicans and Braveheart soundtracks.

Dason

I couldn't remember if we had a general TIL thread - I know we have an R specific one but this is just math related.

Today I learned (or rather discovered) that Euler's formula is a much easier way to derive/remember the "Sine of sums" and "Cosine of sums" trig identities (relevant wikipedia link).

So Euler's formula tells us that

$$e^{i\alpha} = \cos\alpha + i \sin\alpha$$

Now we'll apply that to

$$e^{i(\alpha+\beta)} = e^{i\alpha} e^{i\beta}$$

Now use Euler's formula on both sides

$$\cos(\alpha+\beta) + i \sin(\alpha + \beta) = (\cos\alpha+i \sin\alpha)(\cos\beta + i \sin\beta)$$

Now expand the right hand side

$$\cos(\alpha+\beta) + i \sin(\alpha + \beta) = (\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta)$$

Now to get the identities of interest we just equate the real portion and then the imaginary portion of the equation giving us

$$\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$$
$$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$

I always have an easier time remembering derivations compared to just memorizing formulas. I'm not sure how often anybody here comes across a need for the sum of sines or sum of cosines identities - I know I needed them a lot in my PhD theory courses once we started using characteristic functions to solve everything...

Dragan

Super Moderator
I'm not sure how often anybody here comes across a need for the sum of sines or sum of cosines identities - I know I needed them a lot in my PhD theory courses once we started using characteristic functions to solve everything...
I did, in the area of Order Statistics, when I derived Gini's index of spread in the context of the (standard normal) power method transformation...i.e. because the standard normal cdf isn't available in closed form (and it's being raised to a power in the integral).

TheEcologist

Global Moderator
You know it's going to be a bad day when you want to put on the clothes you wore home from the party last night but you can't find them anywhere ...

spunky

Can't make spagetti
You know it's going to be a bad day when you want to put on the clothes you wore home from the party last night but you can't find them anywhere ...
pictures plz? hlsmith

Less is more. Stay pure. Stay poor.
Were they edible. If so, you just have to wait to get them back.

TheEcologist

Global Moderator
pictures plz? I really really hope there were no pictures taken 