How far, and which way: Instead of 'across and up' (a + bi), describe z by how far it is from the origin and which direction it points.
Polar form
- Modulus r: how far z is from the origin.
- Argument θ: the direction, measured as the angle from the positive real axis.
- Polar form: z = r(cosθ + i sinθ), written short as r cisθ.
r is the length of the line; θ is its angle up from the positive real axis.
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IB-style question — Cartesian to polar
Write z = 1 + √3 i in polar form.
Step by step
- Modulus: r = √(a² + b²).
- Argument: it's in the first quadrant, so θ = arctan(b/a).
- Put r and θ into polar form.
Final answer
2 cis(π/3) = 2(cos 60° + i sin 60°).
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arctan only knows two quadrants: The calculator's arctan(b/a) always lands in quadrant 1 or 4. Sketch the point first — if it's in quadrant 2 or 3, adjust the angle by ±π so θ points the right way.
−√3 + i is in quadrant 2, so its argument is 5π/6 (not the bare arctan).
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IB-style question — a quadrant-2 number
Write z = −√3 + i in polar form.
Step by step
- Modulus.
- Reference angle from arctan(1/√3) = π/6. But the point is in quadrant 2 (left, up).
- Adjust: θ = π − π/6.
- Polar form.
Final answer
2 cis(5π/6).