When do we use euler's equation?

Question

The equation is 'e ^ i x = cos(x) + i sin(x)'

what does this equation actually show? why was it made in the 1st place?

Answer 1

This equation comes up a lot at least for me. I am an electrical engineer and it is used all of the time in signal processing and communications. It is sometimes easier to work with complex numbers than real numbers. Deriving trigonometric identities becomes trivial when you convert the sines and cosines into complex exponentials.

Answer 2

One (somewhat useful) example, is to find the cube roots of -1. Obviously -1 is a cube root of -1, since (-1)^3 = -1, the equation x^3 + 1 = 0 should have three distinct solutions, so what are the other two. Obviously, we could factor x^3 + 1 = (x + 1)(x^2 - x + 1), and use the quadratic formula to find the other two. However, there's another way, using the Euler equation:

Since e ^ ix = cos x + i sin x, then -1 = e ^ (i * pi)

Since x in this case literally refers to an angle around the unit circle, i * pi, 3i * pi, and 5i * pi are all equivalent, so e^(i * pi) = e^(3i * pi) = e^(5i * pi)

Nothing that cuberoot(x) = x ^ (1/3), we can now apply that, to get that the three cube roots of -1 are:

e^(i * pi/3), e^(i * 3pi/3) and e^(e * 5pi/3)

Answer 3

It's more convenient to use the inverse relations:

cos(x) = 1/2i*(e^ix + e^-ix)
sin(x) = 1/2i*(e^ix - e^-ix)

The reason:it's easier to deal with an exponential than with a cos or sin.

Answer 4

You can also make some nice trig identies involving powers of sine and cosine.

Answer 5

Every time when you are working with complex numbers

Answer 6

I tend to avoid using it at all costs