m4230s11hw4.pdf - Math 4230 Assignment 4 due Wednesday...

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Math 4230 Assignment 4, due Wednesday, February 16th.(1) Chapter 3.2, #6: Establish the trigonometric identities sin2(z) + cos2(z) = 1 andsin(z1+z2) = sin(z1) cos(z2) + sin(z2) cos(z1) for the complex sine and cosine.(2) Find all complex numberszsuch thateiz= 4.(3) Ch 3.2 #22: Prove that for anymdistinct complex numbersλ1, λ2, . . . , λm(λi6=λjfori6=j), the functionseλ1z, eλ2z, . . . , eλmzarelinearly independentoverC. In otherwords, show that ifc1eλ1z+c2eλ2z+· · ·+cmeλmz= 0 for allz, thec1=c2=· · ·=cm= 0. [HINT: Proceed by induction onm. In the inductive step, divide by one ofthe exponentials and then take the derivative.](4) Solve the following equations for all possible values of

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