Why does Euler's identity work only in Radians?

[nomadreid]

e^iA = cos A + i*sinA is true iff A is expressed in Radians. Why that particular unit? (Feel free to refer to principles of higher mathematics.)

 

[Deleted member 4993]

e^iA = cos A + i*sinA is true iff A is expressed in Radians. Why that particular unit? (Feel free to refer to principles of higher mathematics.)

Hint:

Following identity is true only in radians:

\(\displaystyle \lim_{\theta \to 0}sin(\theta) \ = \ \theta}\)

Look at the derivation of Euler's identity (or proof of Euler's identity) - for example at:

http://www.chem.mtu.edu/~tbco/cm416/euler_identity.html

 

[soroban]

Hello, nomadreid!

This may not answer your question,
. . but here is an informal explanation for the use of radians.


Many years ago, someone divided the circle into 360 parts,
. . and called them "degrees".

More recently, someone else divided the circle into 400 parts,
. . and called them "gradients".

We could divide the circle into , say, 100 parts,
. . and call them, um,"centians".

You see, anyone can choose an arbitrary number
. . and invent a measure of an angle.

              * * *
           *         *
        *           *   *
       *           *     * r
                  *
      *          * @      *
      *         o * * * * *
      *               r   *
    
       *                 *
        *               *
          *           *
              * * *

Suppose we are given a circle of radius \(\displaystyle r\).
Construct a central angle \(\displaystyle \theta\)
. . so that it intercepts an arc length of exactly \(\displaystyle r.\)
Then \(\displaystyle \theta\) is defined as one radian.

This measure is constant, regardless of the size of the circle.
. . It does not depend on someone's arbitrary constant.

It is a "natural" measure of an angle.


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


A similar situtation . . .


Consider the exponential function: .\(\displaystyle f(x) \:=\:b^x\)
. . where \(\displaystyle b\) is a positive real number \(\displaystyle \ne 1.\)

The deriative is: .\(\displaystyle f'(x) \:=\:b^x\ln b\)


Consider the logarithmic function: .\(\displaystyle f(x) \:=\:\log_bx\)
. . where \(\displaystyle b\) is a positive real number \(\displaystyle \ne 1.\)

The derivative is: .\(\displaystyle f'(x) \:=\:\dfrac{1}{x}\,\dfrac{1}{\ln b}\)


In both cases, since \(\displaystyle b\) is an arbitrary constant,
. . the \(\displaystyle \ln b\) appears as an "adjustment factor".

But if the base is \(\displaystyle e\), the adjustment factor is 1.

. . HHence: .\(\displaystyle \begin{Bmatrix}\frac{d}{dx}(e^x) &=& e^x \\ \\ \frac{d}{dx}(\ln x) &=& \frac{1}{x} \end{Bmatrix}\)


It seems that that ugly decimal: .\(\displaystyle e \,=\, 2.718281828459045...\)
. . arises naturally in mathematics.

And that's why \(\displaystyle e\) is called the base of the natural log.


Afterthought

I suppose \(\displaystyle f(x)\,=\,e^x\) should be called a "natural exponential function".
. . But we don't say that.

When someone mentions "an exponential function",
. . most of us immediately think of \(\displaystyle f(x) \,=\,e^x\), don't we?

 

[nomadreid]

Hint:

Following identity is true only in radians:

\(\displaystyle \lim_{\theta \to 0}sin(\theta) \ = \ \theta}\)

Look at the derivation of Euler's identity (or proof of Euler's identity) - for example at:

http://www.chem.mtu.edu/~tbco/cm416/euler_identity.html

Thanks, but this seems to be begging the question. If you say that e^iA needs to be in radians because the MacLauren expansion of e^iA is in terms of sinA and cosA, and the sinA and cosA has to be in Radians, the question becomes: why do the sinA and cosA have to be in Radians? After all, with the MacLauren expansion we have sin(0) and cos(0), which are 0 and 1 regardless whether we are talking about 0 radians or 0 degrees. Secondly, the proof that the limit of sin x as x goes to 0 = x is essentially taken from Euler's identity, and to say that the limit works only if x is in radians comes from the fact that Euler's identity is in terms of radians. Therefore we end up justifying the obligation of radians in Euler's identity by appealing to the obligation of radians in ....Euler's identity. Could you elucidate where the bedrock is that requires radians?