Today in Complex Variables we learned about the de Moivre formula. So, like I do every time I learn a formula named after some dude, I had to look up the dude to see who he was.
Abraham de Moivre lived from 1667 – 1754 (another one of those long-lived mathematicians) and was friends with Newton, Halley, and Stirling, among others.
Originally from France, he moved to London and, while there, became pretty obsessed with Newton’s newly-published Principia Mathematica and basically memorized the material.
In fact, he took Newton’s binomial theorem and was able to generalize it to the multinomial theorem. This work (plus the fact that he was friends with Newton, I’m sure) got him membership in the Royal Society in 1697.
[And—I have to mention it, I’m sorry—he was one of the dudes on Newton’s little crony committee that was put together to hear the plagiarism charges against Leibniz in 1712.]
De Moivre’s famous formula originated (apparently) from this derivation in 1707:
which he later generalized to this form:
Euler proved it using his own Euler’s formula, so that pretty much cinched it. The reason de Moivre’s formula is so important is because it creates quite a nice connection between complex numbers and trigonometry.
So we’re doing trigonometric integrals in calculus and one of our homework problems is this little dude:
We rushed through this section of the chapter this morning ’cause we’re behind schedule and I’m a little shaky on them (also I’m dumb), so I went to the calc room in Polya to get some help. I showed one of the tutors in there this integral.
I told him how I thought we should start: since (1-cos2x) is the numerator of the half-angle formula for sin2x, we could multiply both sides of the half-angle formula to change (1-cos2x) to 2sin2x and then go from there.
He said he’d never even thought about solving it like that, but when I asked him what the “normal” method would be for this integral, he didn’t know.
So is there another way of solving this?
I’m digging this trig class. It’s only one credit, but I’m really enjoying it. It’s pretty sad that I avoided all math in high school after Algebra II just because of one horrible, horrible math teacher, but considering I used to break out in hives when I walked down the math wing of the building, I don’t know if trig and calc would have been possible for me back then.
I like to think the six trig functions each have their own personalities.
Sine is the smart one. He’s quiet and clever and knows the secrets of the universe.
Cosine is an average kind of dude, but somehow is incredibly lucky and is almost as knowledgeable as Sine just because they hang out with each other.
Tangent is like the quiet, gentle mother figure of the functions.
Cosecant is the super-socially-connected function. He’s the no-nonsense business type who likes to get things done.
Secant is a jerk. He’s super jealous of Cosine and is bitter that his own derivative involves Tangent.
Cotangent is somewhat misanthropic and resents being the inverse of Tangent. He’s not very fun. Even his derivative is a square (HA!).
Yes, this is what goes on in my brain.
I had this dream last night in which I quit school to become a cartoonist. I quickly realize that this is a mistake and draw a series of cartoons to submit to the math department to get back into school. I call the cartoon “Math Quacks” and it stars two ducks names Sine and Cosine. They swim around and eat bread and make really, really, really stupid jokes. They have evil alter-egos named (of course) Cosecant and Secant [I had them mixed up in my head, but I’m pretty sure this is how my subconscious was wanting it to be].
Near the end of the series they make a baby duck called Tangent. Who also makes really, really stupid jokes.
And I think the water they all lived in was called something like Triangle Lake.
This actually might have to happen.