Eulers Formula?

Question

Euler's formula is e^ix=cosx+isinx where e is the base of the natural logarithm , i is the imaginary unit and x is a real number. If x=2 how do u solve e^ix and cosx+isinx so that e^ix=cosx+isinx ?

Answer 1

It doesn't matter what x is, this formula is always true. The fastest way to prove it is this:

Let f(x)=(cos x + i sin x)/e^(ix). Then f'(x)=[(i cos x - sin x) e^(ix) - i e^(ix) (cos x + i sin x)]/ e^(2ix) (by quotient and chain rules). Distributing the i and simplifying, we find that, f'(x)=[(i cos x - sin x) e^(ix) - e^(ix) (i cos x - sin x)]/e^(2ix)=0. Since f'(x)=0, f(x) must be a constant function, so (cos x + i sin x)/e^(ix)=C. Substituting x=0, we have (cos 0 + i sin 0)/e^(0i)=1/1=1, so C=1, and since this is a constant, (cos x + i sin x)/e^(ix)=1 for all x. Therefore cos x+i sin x=e^(ix).

Alternatively, you could use the taylor series expansions of e^x, cos x, and sin x, but you'll just get the same result, regardless of what x is, and that requires you to know what the taylor series of sine and cosine are. I personally find it easier to derive these expansions from euler's formula than to prove the taylor series and use them to derive euler's formula, but whatever floats your boat.

Answer 2

Why could you favor to? Heres a more effective major question... could this characterize an evidence? or merely an illustration? there's a huge difference. Proofs position self assurance in additional effective elementary recommendations which have already been shown and that we settle for. Demonstrations may be carried out with more effective complicated recommendations, recommendations whose foundations are in accordance with what youre attempting to "prepare" yet are taken at face value nonetheless. Eulers formula is an particularly intensive math theory that is determined with the help of calculus and discrete mathematics. How does prepare eulers formula? utilizing endless Maclaurin sequence. How do you prepare the sequence? With calculus. How do you do calculus on trig in case you dont already understand trig? Trig comes first. those proofs do no longer require Eulers formula, and utilizing it hardly ever counts as a rigorous info.

Answer 3

What's to solve? Eulers Formula demonstrates the (fairly deep) link that exists between the exponential functions and the circular functions. It also gives us a very convenient way to write a 2-dimensional vector in a very compact (and easy to use in calculations) exponential form.

You've asked a non-question:

e^(i*2) = cos(2) + i*sin(2)

by definition. Period. There's nothing to solve.

BTW, 'x' is a real number (as you mentioned) but it also represents a real 'value'. In fact, it is exactly the angle Θ in the polar representation of the unit vector (1,Θ).


Doug

Answer 4

that fomula was proved correct by Euler , so it's certainly correct when x=2

Answer 5

Okay smartie pants. It's a tad bit early for this kind of thinking.