Well, this equation is really a consequence of the more general e^ix = cos(x) + isin(x). This, Euler's Formula, enormously simplifies sinusoidal equations. Most common trigonometric identities can be proven in only 3 or 4 steps if you spend 2 of them converting to/from the exponential form, but are far more complicated in the trigonometric form. Many problems in electricity, magnetism, and basic quantum physics would be drastically less wieldy (more unwieldy?) without it.
I don't know of any cases in which it makes things possible, but there are plenty of cases where it makes things practical.
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Can anyone name some of the actual uses of this equation in solving real world problems?
Well, this equation is really a consequence of the more general e^ix = cos(x) + isin(x). This, Euler's Formula, enormously simplifies sinusoidal equations. Most common trigonometric identities can be proven in only 3 or 4 steps if you spend 2 of them converting to/from the exponential form, but are far more complicated in the trigonometric form. Many problems in electricity, magnetism, and basic quantum physics would be drastically less wieldy (more unwieldy?) without it.
I don't know of any cases in which it makes things possible, but there are plenty of cases where it makes things practical.
Sure. In any equation containing e^iπ, you can simplify by substituting -1.
Here is something that comes up in basically every field of engineering:
https://en.wikipedia.org/wiki/Fourier_analysis