# 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

##### Doesn't actually exist
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

##### Omega Contributor
Were they edible. If so, you just have to wait to get them back.