241 ENEE 02*
HOMEWORK ASSIGNMENT 2
Due Mon 02/04
Consider the complex numbers u = ej (/7) , v = ej (6/7) , and w = ej (2/5) (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 u2 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, w2 , w3 , w4 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 + w2 + w3 + w4 = 0 (Do not explicitly and calculate 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
cos
k=1
k N
+ j sin
k N
,
expressing your answer in Cartesian form. (Use the identity 1 + 2 + + n = n(n + 1)/2 to simplify the answer rst.) S 2.2. Show that the expression f () = ej 2 ej + 1 ej + ej 2 , where ranges over [0, 2 ), is real-valued, and obtain an alternative expression for it in terms of sines and/or cosines. S 2.3. Clearly, ej 3 = ej
3
By expanding (cos + j sin )3 and separating real and imaginary parts, obtain two identities: one for cos 3 in terms of powers of cos only, and another for sin 3 in terms of powers of sin only. S 2.4 (more dicult). Consider two complex numbers w and z , where w = 0 is xed and z is variable such that |z | = 1. Show that as z traces out the unit circle, the ratio |z w | |z (1/w)| is constant in value. (Hint : If |z | = 1, then 1/z = z .)