Well, I am into mechanical engineering. So, I have some idea about the applications. In control system, we use complex analysis to know about the stability of any system. In bridge, when the vibration becomes equal to natural frequency, it oscillates with large displacement due to which it may collapse. We may get an idea of the frequency of vibration using complex numbers(FFT, Fast Fourier Transform; Fourier transformation uses complex functions.). Generally while solving problems in the paper, it leads us to an equation containing differential called differential equation. For certain type of diff equations, complex analysis is needed to solve them. Probably, you may have heard about quantum mechanics. In it, the wave function is complex. It is the fundamental physical parameter to approximate various possibilities. Probably, there may not directly use of complex number in daily life. But, it is important especially in deeper research.

The one you talking about is euler relation which connects exponential function with trigonometric using complex number.

e^ix = cos x + isin x