Euler's Formula: Learn Trigonometry & Answers to 3 Questions

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e^i^\theta = cos \theta + i sin \theta

z = x + i y = r (cos \theta + i sin \theta ) = re^i^\theta

I have 3 questions about these kinds of formulae;\alpha \lambda \phi \alpha : Does anyone have any links that will teach me how to use trigonometry using these kinds of formulae and learn all about them in general?

\beta \eta \tau \alpha : is the "e" the exponential e and if so, how can a number such as 2.71828... be obtained from trig functions who must be squared to get to 1, let alone 2.71828...

\Gamma \alpha \mu \mu \alpha : I've heard it said that all of trigonometry can be summed up using these kinds of formulae (Leonard Susskind), well why aren't we taught how to look at trig in this way?

 

[n!kofeyn]

http://books.google.com/books?id=og...omplex analysis&pg=PA10#v=onepage&q=&f=false" by Tristan Needham. I don't know what your level of math is, but this might be a good place to start. You might go back to the beginning of the book to get his geometric notation down.

I've watched a few of Susskind's lectures online, which put me to sleep, and I never felt like he was even teaching anything. He just talked a lot.

 

[keniwas]

I don't know that Needham is going to be a good place to start... while he covers the basics, if you really don't know anything on the topic its not going to be very smooth reading in my opinion. Though if your goal is to do trig using exponentials then he has a brilliant presentation on the topic. (i.e. he shows you quite easily how all trig identities can be derived almost trivially using exponential forms)

If you are looking for where the formula comes from, any standard text on calculus will include a section on taylor expansions, and will certainly include a section on the power series of Sine and Cosine, from which is trivial to derive Euler's formula.

the 'e' is the expnential e. It is obtained in a number of different ways (generally the limit of a seq like
<br /> \lim_{n\to\infty}\left(1+1/n\right)^n<br />

or somthing similar).
, however to your question as to how you can get it from sines and cosines, the answer is in the power series expansion of each of those two functions. Its actually easier if you expand the exponential form and factor out the series for i*sin(x) and cos(x).

As to why you aren't taught trig this way, it's probably becuase it generally supposes knowledge in calculus and complex variables, which at the time most people take Trig they aren't familiar with (its generally a prereq for calculus). That said, calculus isn't required at all to use it to solve trig problems, so long as you are ok accepting Euler's formula as a fact and don't care about why its true.

 

[n!kofeyn]

That's exactly the section that I linked to, and he also discusses the Taylor series of sine and cosine. When I linked directly to the Google Books chapter on Euler's formula, I only meant for the original poster to take a look at those first thirty pages.

 

[emyt]

e^i^\theta = cos \theta + i sin \theta

z = x + i y = r (cos \theta + i sin \theta ) = re^i^\theta

I have 3 questions about these kinds of formulae;


\alpha \lambda \phi \alpha : Does anyone have any links that will teach me how to use trigonometry using these kinds of formulae and learn all about them in general?

\beta \eta \tau \alpha : is the "e" the exponential e and if so, how can a number such as 2.71828... be obtained from trig functions who must be squared to get to 1, let alone 2.71828...

\Gamma \alpha \mu \mu \alpha : I've heard it said that all of trigonometry can be summed up using these kinds of formulae (Leonard Susskind), well why aren't we taught how to look at trig in this way?

we aren't taught trig that way because it isn't "necessary" for the use of trigonometric functions

 

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Brilliant. Thanks a lot I've only skimmed now as I have no time but I'll be back should I have any problems. spasibo, merci, gratias vobis ago