/sci/ - Science & Math

>>6976651
You forgot Re() in front of those sums

Also, they are defined like that

>>6976651

They are.

Pre-cal classes use the grade school definitions because they're more useful for the trig that's covered in the class, and series are usually a later topic.

ok. I think schools should introduce complex numbers way earlier. And stop calling them imaginary

I feel like i^n is needlessly abused in many of these situations but I can think of a good alternative besides reducing the series.

>>6976668
complex numbers = real component + imaginary component

Agreed OP, fuck learning the sin and cosine as trig functions, let's teach power series to them 8th graders

>>6976685
This turned out to be not nearly as clean our insightful as I imagined it would be.

>>6976685
This turned out to be not nearly as clean or insightful as I imagined it would be.

>>6976651
Because infinity doesn't exist.

>>6976651
Either use:
<span class="math"> sin(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}[/spoiler]

<span class="math"> cos(x) = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}[/spoiler]

or use: <span class="math"> e^(z) = \sum_{n=0}^\infty \frac{z^n}{n!}[/spoiler]
and <span class="math"> cos(x) = \frac{e^{ix} + e^{-ix}}{2}, sin(x) = \frac{e^{ix} - e^{-ix}}{2i}[/spoiler]

>>6976651

This is what I've always thought. I know everyone here is insisting that it is defined like this, but it should at least be noted to school children that it can be expressed like this. Just so that these functions don't seem arbitary, pulled out of thin air.

I learned these series leafing through a formula book, and to later learn that these were discovered in 14th century India.

https://en.wikipedia.org/wiki/Madhava_series

>>6976651
Because that's objecively wrong you retard

Those power series aren't sin and cos at all.

>>6976875
>mind=blown

>>6976651
>>6976760
>>6976805
When will I be able to understand this sort of stuff? I just finished Precalculus Algebra and Trigonometry.

>>6976969
4chan is for >18 yo

>>6976972
I'm taking them in college because I didn't understand the educational system in high school. Blame absent parents or shitty teachers or America if you want.

>>6976800
Just as numbers don't.

>>6976651
You can adjust by taking real part of the sums and some other minor modifications and you should be able to use i^n.

>>6976705
What about other impossible values, like one over zero? Why doesn't it get it's own "imaginary" variable? 1/0 and the root of negative one are both undefined.

>>6976996
The imaginary unit is not sqrt(-1)
It is defined as the solution for x^2 = -1.
There's a difference.

>>6976805
The second set of definitions there literally follows from the babby definitions your so desperate to distance yourself from.

>>6976996
there is no consistent definition for 1/0 that could be viable in mathematics that I know of

>>6976897
Wow why are you such a dick?

>>6977063
There are algebraic structures where you can divide by 0, and they actually have a somewhat interesting projective interpretation. Nothing too important or useful though

>>6976969
You'll understand it when you start working on it.

>>6976651
You can do it this way (if you correct the formulas!) but usually sin() and cos() are introduced in trigonometry, where the use of the unit circle makes their usefulness and geometric interpretation clear. For example, it's not clear from the series definition that sin^2+cos^2=1 or that they are periodic with period 2 pi, or that they can be used to describe ratios of the lengths of sides of right triangles.

>>6976657
We dont need to use Re() because we're dealing with imaginary numbers here. They're not real, so we can discard them.
>>6977150
The series definition is perfect plug and chug for all real numbers. We don't need to define anything else. Teaching the underlying properties is of course important.

>>6977055
The difference is that the second set are accurate definitions. The definitions in the original image are crappy, since they imply that sin or cos of a real number can give a complex number, which is obviously wrong. They need a Re(series) operator just to make them correct, and then they will give the wrong answer when you try so take sin or cos of a complex number--which will indeed be complex. And even if it does work for some values of x, it doesn't mean that form will be useful.

By contrast, writing sin and cos in terms of e^(ix) is tremendously useful. And the form written in >>6976805 will give the right answer without needing to take the real part: the functions are real-valued for real x and complex-valued for complex x by construction.

>>6976651
In order to define the trigonometric functions that way, you need all of calculus and analysis first. Which is dumb.

>>6976976
But don't blame the precious innocent child! You could never be at fault for never picking up a goddamn book, or God forbid, looking something up on the internet like the rest of us!

>>6977177
>they're not real, so we can discard them
It doesn't work that way, Dubs. If defined this way, plugging in reals into sin/cos gives complex numbers (generally). This fucks up formulas involving other complex numbers.

Suppose that you multiply (a+ib)(c+id): the result is (ac-bd)+i(ad+bc), not ac (assuming b,d are discarded, since they're "not real"). Real and imaginary parts are interwoven.

>>6976651
If you add Re() around them, you get equivalent definitions. It's clear why they're equivalent with the commonly used ones, so feel free to use them when it simplifies your calculations, or use the common ones when they lead to better calculations instead. It's not the definition that matters, it's how you use it.

>>6977291
This is correct and very well-said.

>>6977050
i is definded as sqrt(-1)

x^2=-1 has two solutions: i and -i

>>6976651
WTF? Sin and Cos are real valued functions. What you wrote makes no sense.

>>6977050
You're confused. You're saying there are two imaginary units. Nonsense.

>>6977316
except he's doing that right now you shit heap

>>6977313
You don't need all of analysis. Lebesgue integration? Not needed. Epsilon delta definition of a limit? Not needed.

You don't need calculus either. No integrals or derivatives required to take the limit of a sequence.

Especially the last line of equations in >>6976805 all you need to know is rules of exponents and imaginary numbers to start deriving trigonometric identities.

Although, to be honest, screw trigonometric functions. We should just be teaching kids dot products anyway. Easier to calculate by hand, and better for most computer simulations and scientific applications.

>>6978376
How do you rigorously define the limit of a sequence without Epsilon-delta (or more like epsilon-N here)?

>>6978263
just take the real part. It's just a somewhat more convenient way of writing (-1)^n

>>6978382
>protip : you don't

>>6978382
well, if you use it for continuous functions, point-set topology is a nice and rigorous 'alternative'. But if you really want sequences, you can consider nets (which is a generalization of sequences), for which convergence is defined with point-set topological notions.

>>6977291
yep. Also using the exponential makes everything more simple and homogenous, since you're using a well understood and intuitive function instead of a fuckton of trigonometric abortions that will make your integrals a headache

>>6978408

>i^n is more convenient than writing (-1)^n

autism

>>6978359
Not the guy you responded to, but we can perfectly well blame him for not being motivated sooner. I was reading Shankar in my Junior year of HS. I didn't need anyone to motivate me, so why should he? He should take some responsibility.