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While I am studying trigonometric functions from $\mathbb{R}$ to $\mathbb{R}$, I thought their arguments are expressed in radians. Thus it has elegant series expansion.

For example, seeing real sine function $sin x$, we can think $x$ as $x$ radian, not $x$ degree or just real number $x.$

How about the complex function sin z? Here $z\in \mathbb{C}.$ Is it possible to think $a+ib$ as $a$ radian +$i b$ radian like real sine function?

Are their any relation between complex numbers and radians?

One more question: $sin x$ has a Taylor expansion. If we think $x$ as $x$ radian, is $sin x$ real number or $sin x$ radian? For simplicity, if $x$ radian, what is $x^2$ ? Is it real number or $x^2$ radian^2?

Would you give any comment for it? Thanks in advance!

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  • $\begingroup$ Do you mean "what does complex angle mean?" Or "what is the relation between the complex number $z=a+ib$ to radians?"? $\endgroup$ – ℋolo Mar 13 '18 at 13:36
  • $\begingroup$ @Holo I’d like to know the relation between the complex numbers and radians. Is it possible to consider complex number $a+i b$ as $a radian +i b radian$? $\endgroup$ – 04170706 Mar 13 '18 at 13:39
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First, what is the connection between angles and complex numbers?

Let's say we have a complex number: $z=x+iy$, where $x,y$ are real and $i$ is the imaginary unit.

When we talk about real numbers we can put those numbers on the number line, similarly we can put complex numbers on the complex plane. We will call the 'x'-axis $\Re$, from real, and the imaginary part $\Im$, from imaginary. Our number $z$ will be on the point $(x,y)$(see picture) enter image description here

In the picture we can see that we can also look at the point as a line from $(0,0)$ to $(x,y)$, so we may write the number $z$ as $r(\cos(\theta)+i\sin(\theta))$ where $\theta$ is the angle with the real axis and $r=\sqrt{x^2+y^2}$, we now have new way to write complex numbers: $(r,\theta)$ this is the polar form of the complex number.


If you want connection with specificily radians than we need to look a little deeper, Euler formula is $re^{i\theta}=r(\cos(\theta)+i\sin(\theta))$, this formula can be proved using a lot of different methods, you can search on the internet or ask in a comment of you like to know it. This formula is something originally derived from Taylor series of sine and cosine, the series is a way to express sine and cosine using sum of polynomials and this series is using radians and not degrees.

Also everywhere we see angles we see circle, and you already should know what is the connection between radians and circle.


Edit

When we have $\sin x$ we view $x$ as an angle we don't need to distinguish between Measurement Units, but angles are defined to be between lines, so in normal conditions there is no such thing as "complex angle" but we still generalize the trigo functions to the complex plane(you can easily find the generalize form online). There is no easy way to tell exactly what we mean by $\sin(i)$ because it there is no angle of size $i$, but we can say, for example that $\sin(a+ib)$ is some kind of combination of the angle $a$ and $b$:$\sin(a+ib)=\sin(a)\cosh(b)+i\cos(a)\sinh(b)$, this has a little more sense in the way we see it.


We can also look at it more straight forward, we can easily define the unit circle in $\Bbb R^2$, but what about $\Bbb C^2$, although we can change it without loosing generalization to $\Bbb R^4$ we don't always want to, saying that angles can be complex gives us the very simple form of $(\cos(z),\sin(z))$ to be the complex unit circle of $\Bbb C^2$.

I am not very knowledgeable on the subject of complex angles but there is very nice paper exactly about this: This is an 8 pages long paper by Richard Hammack


On to the second question:

What is radians? We define $\theta=\frac{\text{arc}}{\text{radius}}$

What is arc? We measure arc using length, so metre or something along those lines, but we measure radius using the same units, so $\theta=\frac{\text{arc}_{\text{unit length}}}{\text{radius}_{\text{unit length}}}$ so the unit of radians is dimensionless unit

So what can we understand from this?

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  • $\begingroup$ Thank you for your answer. For real sine function, $sin x$, we can think $x$ as $x$ radian, not $x$ degree or just real number $x$. I’d like to know what we say about complex function $sin z$. $\endgroup$ – 04170706 Mar 13 '18 at 14:10
  • $\begingroup$ @0706 see edit, I hope this is what you meant $\endgroup$ – ℋolo Mar 13 '18 at 14:31
  • $\begingroup$ @0706 I added answer also to the second question $\endgroup$ – ℋolo Mar 13 '18 at 15:46
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The usual functions $\sin, \cos$ in calculus are functions from $\mathbb{R} $ to $\mathbb{R} $ and radian/angles etc are just a way to look at these functions. Important point is that you don't need to know about radians to describe these functions. And further these functions can be extended to complex arguments also making them a function from $\mathbb{C} $ to $\mathbb {C} $.

Coming back to measurement of angles, just as with some set of points in a plane we can associate a number called its area it is possible to associate a number to an angle formed by two rays in plane emanating from a single point. This is what one calls radian measure, but the scenario is different from the way we use units in physics to measure physical quantities. Here the measurement is just a number whether we are measuring length, or area or angle or volume.

Further the standard way of measuring angles amounts to associating the number $\pi/2$ with a right angle. Now who likes an irrational number associated with a right angle? So some people just adopted a different convention and associated the number 90 to a right angle. To distinguish between two standards above it is necessary to name them and thus first standard came to be known as radian measure and the second standard was named the degree measure.

The definitions and properties of circular functions take a simple form if the radian standard is used and that's the default in calculus /analysis. But above all one must understand that as far as mathematics is concerned these measurements are just real numbers and nothing more. The extension to complex numbers is not related to any measurement of angles and one should not try to think in that manner.

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In terms of complex valued trigonometirc functions, thier Taylor expansions are as follows

$$\cos(z) = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - ... \frac{z^{2n}}{(2n)!} + ... = \sum_{n=0}^{\infty} \frac{(-1)^nz^{2n}}{(2n)!}$$

$$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - ... \frac{z^{2n+1}}{(2n+1)!} + ... = \sum_{n=0}^{\infty} \frac{(-1)^nz^{2n+1}}{(2n+1)!}$$

Now one can also write $$\sin z = \frac{e^{iz}-e^{-iz}}{2i}$$ Where the complex exponential form of a complex number $$z= r e^{i \theta}$$ is such that $r$ is the modulus of the complex number and $\theta$ is the angle in radians.

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  • $\begingroup$ Thanks for your answer. It seems that radian is related to complex numbers, but I can’t see it exactly. I’d like to know the relation between real function and complex function for radians. Do you have any idea? $\endgroup$ – 04170706 Mar 13 '18 at 12:39
  • $\begingroup$ Well if I understand you correctly, $360$ degrees $= 2 \pi$ radians so the relationship is between these two elements really. $\endgroup$ – Kevin Mar 13 '18 at 12:52

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