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During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting:
$ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $
$ \sin ( 1 + i ) \approx 1.30 + 0.63 i $
Does this mean that there is such a thing as "imaginary" or "complex" angles, and if they do, what practical use do they serve?
These values are relying on the generalization of the sine function as $$ \sin x = \dfrac{ e^{ i x } - e^{ - ix } }{2i}. $$ Clearly, when $x$ is complex it cannot be interpreted geometrically as an angle; however, generalized in this way, $\sin x$ becomes a holomorphic function, which is nice for a variety of other reasons. Practically, "imaginary angles" have some applications in physics. For instance, in optics, when a light ray hits a surface such as glass, Snell's law tells you the angle of the refracted beam, Fresnel's equations tell you the amplitudes of reflected and transmitted waves at an interface in terms of that angle. If the incidence angle is very oblique when traveling from glass into air, there will be no refracted beam: the phenomenon is called total internal reflection. However, if you try to solve for the angle using Snell's law, you will get an imaginary angle. Plugging this into the Fresnel equations gives you the 100% reflectance observed in practice, along with an exponentially decaying "beam" that travels a slight distance into the air. This is called the evanescent wave and is important for various applications in optics.
See A geometric view of complex trigonometric functions by Richard Hammack in The College Mathematics Journal, May 2007; 38, 3 pages 210-217, for a much more thorough discussion of complex angle as hyperbolic angles.
Yes, the sine function can be defined meaningfully on the whole complex plane as is explained here.
Whether or not you want to call a complex argument for the sine function an angle is a different question and, I suppose, a matter of taste.
The Optics example can go farther:
Snell's Law states that $n_{0} \, \sin(\theta_{0})= n_{1} \, \sin(\theta_{1})$. For absorbing materials (or conductive like Au or Ag) the "$n_{1}$" is a complex number. This requires that $\theta_{1}$ also be complex.
A fundamental equation of trigonometry is $x^2+y^2 = 1$, where $x$ is the "adjacent side" and $y$ the "opposite side".
If you experiment plot $f(x)$ out of the real domain - for example to $x=1.5$ you obtain $y$ imaginary - you will get an imaginary shape situated in a plane perpendicular to the plane $x,y$ and containing the $x$-axis. This shape is a hyperbola.
So you have two planes, one for the circle, and one for the hyperbola. The "$z$-axis" (imaginary) where the hyperbola is plotted correspond to the "$\sinh$" and $x$ is the "$\cosh$" once the $R = 1$. Note that the $\sinh$ is situated in a plane $90$ degrees of the $x,y$-plane. Observe that $$\sin iy = i \sinh y$$ is in accord with was explained above.
The geometric interpretation is easy.
It's valuable remember that the angle of a circumference could be measured by the double of the area of the sector. The hyperbolic angle could be measured by the double of area limited by the radius and the arc of hyperbola. See Wikipedia.
Many areas in physics use what's called the Euler formula, which relates trigonometric functions to complex exponentials. This case, however does not actually admits complex angles: The arguement of the complex exponential is $e^{i\theta}$, which is interpted as either real trigonometric function $\cos(\theta)$ or $\sin(\theta)$.
On the other hand, a complex angle is a whole different story. I'll be giving now a qualitative physical explanation for why complex angles cannot be physical.
Let's assume that we have a certain physical system, say a pendulum - in which the energy depends on an angle $\theta$ by the equation $E=-mgL\cos(\theta)$. If we now allow the system to have hypothetical complex angle: $E=-mgL\cos(i\theta)$. Due to the identity $\cos(i\theta)=\cosh(\theta)$ we have an energy $E=-mgL\cosh(\theta)$. As we all know, this means that the energy is not bounded from below (as $\cosh(\theta)$ increase exponentially as $\theta$ tends to infinity) - and this is a serius break of the laws of physics. If it was true, then the system would tend toward the lowest energy state, and the conservation of energy demand it to release energy to the enviroment as the system minimizes it's energy. Thus creating infinite amount of energy out of the complex plane.
Here's a more literal interpretation.
Yes, you can actually create geometric spaces in which angles can be imaginary and even complex.
The trick is to note that the most general definition of angle is staged within the framework of inner product spaces: vector spaces which have an operation called the "inner product" that in effect defines how a "dot product" works for that type of vector. This definition is based on how that, for vectors in Euclidean $n$-dimensional space, we can define the angle between them via
$$\theta(\mathbf{v}, \mathbf{w}) := \cos^{-1}\left(\frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| |\mathbf{w}|}\right)$$
which, in turn, is derived from the "geometric dot product"
$$\mathbf{v} \cdot \mathbf{w} = |\mathbf{v}| |\mathbf{w}| \cos \theta$$
where $\theta$ is the angle between the vectors, but we now go the other way and take the dot product as a primitive operation, not length and angle, and we use that to define length and angle, which are concepts that don't exist in an ordinary vector space. (Thus, we should note also that the length of $|\mathbf{v}|$ is then defined by, as one might think, $|\mathbf{v}| := \sqrt{\mathbf{v} \cdot \mathbf{v}}$.)
To see how this framework works, note that for real, say 2-dimensional, Euclidean vector space, we have that vectors look like
$$\mathbf{v} = \langle v_x, v_y \rangle$$
so that
$$\mathbf{v} \cdot \mathbf{w} = v_x w_x + v_y w_y$$
and from which we get the usual angle between two points in 2-dimensional space.
However, because we are talking more general vector spaces, there is no need for the scalar components of a vector to be real numbers only: we can - and do! - just as well take them as complex numbers. Of course, if we have vectors of two complex numbers, that is effectively like having four real-number dimensions, but we are defining a new kind of geometry on that. In particular, if we now have two-dimensional vectors $\mathbf{v}$ with the same form as given, only components $v_x$ and $v_y$ are complex, so that the space has two complex dimensions (and so four real dimensions), we can define an inner product like
$$\mathbf{v} \cdot \mathbf{w} := v_x \bar{w}_x + v_y \bar{w}_y$$
where you note we take the conjugate: this is to ensure that in various ways the dot product is "nice" to the structure of the complex numbers. Moreover, it ensures that the length of a vector continues to remain real (exercise: check this by considering what $\mathbf{v} \cdot \mathbf{v}$ is and then think of a certain fundamental property of the complex numbers).
So then if we do this, and we return to our definition of $\theta(\mathbf{v}, \mathbf{w})$, we can see that while the denominator of what's inside $\cos^{-1}$ is just lengths and so will be real, the numerator, which is a bare inner product, has no need to be real at all, and so we can in general get an inverse cosine of a complex number and so find vectors with an angle which is complex.
