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ENEE 241 02* HOMEWORK ASSIGNMENT 2 Due Mon 02/04
Consider the complex numbers
u = e
j ( π/ 7)
, v = e
j (6 π/ 7)
, and w = e
j (2 π/ 5)
(i) (2 pts.) Plot the three numbers on the complex plane.
(ii) (3 pts.) Plot the straight line described by the equation | z - u | = | z - v | (where z is a variable
point) on the complex plane.
(iii) (4 pts.) Plot the circle described by the equation | z - u - u
2
v | = | w | . Which of the three
points (i.e., u , v , and w ) does the circle pass through?
(iv) (4 pts.) Each of the expressions v + v
13
and v - v
13
can be evaluated using a single trigonometric
function (either sine or cosine). Why is this so? ( Hint . What other power of v does v
13
also equal?
Plot v
13
on the complex plane if necessary.)
(v) (4 pts.) Which polynomial of degree n = 5 has w , w
2
, w
3
, w
4
as its roots?
(vi) (3 pts.) Based on your (hopefully) correct answer to (v) above, use the geometric sum formula
to show that
1 + w + w
2
+ w
3
+ w
4
= 0
(Do not explicitly calculate and add all these numbers.)
Solved Examples
S 2.1 ( P 1.10 in textbook). Let N be an arbitrary positive integer. Evaluate the product
N - 1
Y
k =1
cos