What are some uses of Euler's formula?

Answer 1

There are two different formulas, which are commonly referred to as Euler's formula. Links are posted below.

One has to do with solid geometry, which states that F + V - E = 2, which means (number of Faces) + (number of edges) - (number of vertices) equals 2. This may be useful if you want to build a structure of a certain shape (maybe a dodecahedron, which is a 12 sided shape) and help to figure how much building materials you will need.

The other Euler's formula is used with complex numbers, which states e^(i*Θ) = cos(Θ) + i*sin(Θ), where i is the imaginary unit. Note that (Θ) must be expressed in radians to work properly.

So thinking about triangles and the unit circle, any complex number can be graphically represented by the real portion on the horizontal, and the imaginary portion on the vertical. One advantage, is complex numbers are easier to multiply, divide, and raise to powers if they are represented as e^(iΘ) rather than a + bi.

Another use of this is figuring the converse trig functions:

cos(Θ) = [e^(iΘ) + e^(-iΘ)] / 2, and sin(Θ) = [e^(iΘ) - e^(-iΘ)] / (2i)

If you can remember these, you can figure any trig identity that you may need to use, from these two.