## Saturday, May 20, 2006

### 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 = cos α + i sin (α)

(2) So, we see that:

(e)a = ei*a*α

(3) Finally,

ei(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 = cos α + i sin α

e = cos β + i sin β

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

(e)(e) = eiα + iβ = ei(α + β)

(3) Finally,

ei(α + β) = cos(α + β) + i sin(α + β)

QED