Question

Simplify these numbers: (a) e 4jπ (b) e 2015jπ

2. (5 pts) Simplify these numbers: (a) edim (b) e2015jm 3. (10 pts) Logarithms: Simplify the following as much as possible, without using a calculator i slow the various steps .

Answer 1

e^j 4 pi We know that e^j theta = cos theta + j sin theta e^j 4 pi = cos 4 pi + j sin 4 pi = > e^j 4 pi = 1 + j0 = > e^j 4 pi = 1 e^j 2015 pi We know that e^j theta = cos theta + j sin theta = > e^j 2015 pi = cos(2015 pi) + j sin(20 15 pi) = > e^j 2015 pi = cos(pi + 2014 pi) + j sin(pi + 2015 pi) n epsilon Integer Let 2014 pi = 2n pi = > e^j 2015 pi = cos(pi + 2 n pi) + j sin(pi + 2 n pi) = > e^j 2015 pi = cos pi + j sin(pi) = > e^j 2015 pi = -1 Using sin(theta + 2n pi) = sin theta cos(theta + 2 n pi) = cos theta