Name ___________________
Worksheet 12.2
De Moivre’s Theorem
1. Consider the formula z  r  (cos   i sin  )  r  cis
(a) Find z2 in terms of r and θ
(b) Find z3 in terms of r and θ
(c) Show that zn = rn(cos(nθ) + i sin(nθ)) This is called De Moivre’s Theorem.
2. Use the result of part 1 to solve the equation z3=1
by noting that 1  1cis0  1cis360  1cis2  etc. and
using the result from part 1.( remember that an
equation like sin = 1 has infinitely many solutions)
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3. Graph the three solutions to 2 on the complex
plane
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4. Repeat the process in 2 and 3 with z4=1
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5. What do you notice about the pictures you have
created?
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6. Try to draw the solutions of z5=1 without any
calculations and explain why you think your drawing is
correct.
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7. Find all possible solutions to following equations.
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(a) z 2  9
(b) z 6  1
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(c) z 8  1
(d) z 3  i
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(e) z 4  1
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(f) z 6  64
8. Find i