Polar (modulus-argument) form
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Flip to reveal answersWhat is polar (modulus-argument) form?
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All 8 Flashcards — Polar (modulus-argument) form
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Question
What is polar (modulus-argument) form?
Answer
z = r(cosθ + i sinθ) = r cisθ, where r = |z| (modulus) and θ = arg z (argument).
Question
How do you find the modulus and argument from a + bi?
Answer
r = √(a² + b²); θ = arctan(b/a), then adjust for the quadrant.
Question
How do you convert polar back to Cartesian?
Answer
a = r cosθ and b = r sinθ, then write a + bi.
Question
Why must you check the quadrant for the argument?
Answer
arctan(b/a) only returns quadrant 1 or 4 angles; quadrant 2 or 3 points need ±π added.
Question
Write 1 + √3 i in polar form.
Answer
r = 2, θ = π/3, so 2 cis(π/3).
Question
Write −√3 + i in polar form.
Answer
r = 2; quadrant 2 so θ = π − π/6 = 5π/6; 2 cis(5π/6).
Question
Convert 2 cis(π/6) to Cartesian.
Answer
a = 2cos30° = √3, b = 2sin30° = 1, so √3 + i.
Question
What does the argument θ mean geometrically?
Answer
The angle the line from 0 to z makes with the positive real axis.
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Full study notes for Polar (modulus-argument) form
Topic 1.13 hub
Complex numbers: polar (HL only)
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