# Do “imaginary” and “complex” angles exist?

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?

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the classical trigonometric functions defined on the real line with values in $[-1,1]$ admit holomorphic extensions with values in $\mathbb{C}$. The range is no longer $[-1,1]$, though, which kind of explains the result for $\sin^{-1}(2)$ I doubt that the output of the corresponding inverse functions still admits an interpretation as an angle. –  user20266 Jan 3 '12 at 19:03

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.

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I was not aware of this generalization. This is so beautiful! –  Avi Jan 4 '12 at 2:34
+1 for the application to optics! Very nice. –  Chris Taylor Feb 22 '12 at 18:54