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!
Best Answer
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)
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?