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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? | |||||||
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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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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. | |||
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