Some Amazing Applications of Euler’s Formula

Euler’s simple formula has so many cool applications in trigonometry

Euler’s formula is a simple expression that links complex numbers with trigonometry. It goes like this:

cosθ is thus the real part of the identity and sinθ is the imaginary part.

Using Euler’s formula, it’s pretty straightforward to express standard trig ratios in terms of complex exponents, for example, noting that cos(-θ) = cosθ and sin(-θ) = -sinθ, we can say that

and

Deriving well-known trigonometric identities

This way of expressing standard trig ratios offers us the possibility to use Euler’s formula to prove certain standard trig identities, for example:

Rearranging this we have:

Using an alternative manipulation we can determine:

So that:

Subtracting or adding these two identities tells us the well-known high school trig identities: