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Examples a z1 2 2i z2 3i b formula for zn de moivres

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Unformatted text preview: n in Polar Coordinates Finally, since z1 = z1z2 -1, we have: z2 z1 r1 = [ cos(ø1-ø2) + i sin(ø1-ø2)] z2 r2 That is, we divide the magnitudes and subtract the arguments. Examples (a) z1 = -2 + 2i, z2 = 3i (b) Formula for zn De Moivre's formula zn = rn(cos nø + i sin nø) In words, to take the nth power, we take the nth power of the magnitude and multiply the argument by n. Examples Powers of unit complex numbers. 9. nth Roots of Complex Numbers Write z = r(cos(ø+2kπ) + i sin(ø+2kπ)), even though different values of k give the same answer. Then z1/n = r1/n[ cos(ø/n+2kπ/n) + i sin(ø/n+2kπ/n)] Note that we get different answers for k = 0, 1, 2,..., n-1. Thus there are n distinct nth roots of z. Examples 4 (a) i (b) 4i (c) Solve z2 - (5+i)z + 8 + i = 0 (d) nth roots of unity: Since 1 = cos0 + isin0, the distinct nth roots of unity are: çk = cos(2kπ/n) + i sin(2kπ/n), (k = 0, 1, 2, ... , n-1) More examples In class. 10. Exponential Notation We know what e raised to a real number is. We now define what e raised to an imaginary number is: Definition: eiø = cos ø + i sin ø. Thus, the typical complex number is Exponential Form of a Complex Number reiø = r[cos...
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