Moduli and principal arguments in quadrants and on axes

Moduli and principal arguments in quadrants and on axes

IB Year 6 | Grade 12
12 marks

Find the modulus and principal argument of each of the following complex numbers. Give arguments in radians in the interval \(-\pi<\arg z\leq\pi\), using exact values or inverse tangent notation.

  1. \(\sqrt{3}+3\mathrm{i}\)[2]
  2. \(5-12\mathrm{i}\)[2]
  3. \(-2\sqrt{3}+2\mathrm{i}\)[2]
  4. \(-3-3\mathrm{i}\)[2]
  5. \(-7\)[2]
  6. \(-6\mathrm{i}\)[2]

Solution:

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Answer:(i) \(|z|=2\sqrt{3},\ \arg z=\dfrac{\pi}{3}\) (ii) \(|z|=13,\ \arg z=-\tan^{-1}\!\left(\dfrac{12}{5}\right)\) (iii) \(|z|=4,\ \arg z=\dfrac{5\pi}{6}\) (iv) \(|z|=3\sqrt{2},\ \arg z=-\dfrac{3\pi}{4}\) (v) \(|z|=7,\ \arg z=\pi\) (vi) \(|z|=6,\ \arg z=-\dfrac{\pi}{2}\)

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