Properties of cos θ + i sin θ

In today's blog, I will go over some basic trigonometric properties that I use in my proof for the Fundamental Theorem of Algebra.

Lemma 1: [(cos α + i sin α) ]^a = cos (a*α) + i sin (a*α)

Proof:

(1) From Euler's Formula, we know that:

e^iα = cos α + i sin (α)

(2) So, we see that:

(e^iα)^a = e^i*a*α

(3) Finally,

e^i(a*α) = cos (a*α) + i sin (a*α)

QED

Lemma 2: (cos α + i sin α)(cos β + i sin β) = cos(α + β) + i sin(α + β)

Proof:

(1) Again, using Euler's Formula, we have:

e^iα = cos α + i sin α

e^iβ = cos β + i sin β

(2) So multiplying these two values together gives us [see here if a review of exponents are needed]:

(e^iα)(e^iβ) = e^{iα + iβ} = e^{i(α + β)}

(3) Finally,

e^{i(α + β)} = cos(α + β) + i sin(α + β)

QED