hwB02 - + j sin k N , expressing your answer in Cartesian...

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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 ± N
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Unformatted text preview: + 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 ( ) = e-j 2 -e-j + 1-e j + e j 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, e j 3 = e j 3 By expanding (cos + j sin ) 3 and separating real and imaginary parts, obtain two identities: one for cos3 in terms of powers of cos only, and another for sin3 in terms of powers of sin only. S 2.4 (more dicult). Consider two complex numbers w and z , where w 6 = 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 * .)...
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This note was uploaded on 03/21/2010 for the course ENEE 241 taught by Professor Staff during the Spring '08 term at Maryland.

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