assig11

# assig11 - z ∈ C ² ² | z | | z-4 | = 8 ³ 5 For real...

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MATH 135 Algebra, Assignment 11 Due: Wed Dec 2, 8:30 am 1: Express each of the following complex numbers in cartesian form. (a) (2 + i )(3 + 2 i ) - (5 + 3 i ) (b) 1 ( 2 + i )(1 + i 2) (c) (1 + 2 i ) 3 (3 + i ) 2 2: Solve each of the following equations for z C . Express your answers in cartesian form. (a) z + 1 z + i = 3 + i (b) z 2 = z (c) z 2 = 4 + 3 i 3: Solve the following pairs of equations for z, w C . (a) z + i w = 2 i z + 2 w = 3 (b) i z + (1 + i ) w = - 3 + i (2 + i ) z + (3 - 2 i ) w = 4 i 4: Draw a picture of each of the following subsets of the plane. (a) ± z C ² ² 1 < | z - 1 | ≤ 5 ³ (b) ± z C ² ² | z - 2 i | = | z - 4 | ³ (c) ±
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Unformatted text preview: z ∈ C ² ² | z | + | z-4 | = 8 ³ 5: For real numbers x and y , we deﬁne e x + iy = e x cos y + i e x sin y . For a complex number z , we deﬁne cos z = e iz + e-iz 2 and sin z = e iz-e-iz 2 i . (a) Show that for all z, w ∈ C we have e z + w = e z e w . (b) Show that for all z, w ∈ C we have sin( z + w ) = sin z cos w + cos z sin w . (c) Solve e z = 1 + i √ 3 for z ∈ C . (d) Solve sin z = i for z ∈ C ....
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