IB Maths AA HL Ahl 1 14 Hl Complex Roots and De Moivres Theorem Questions

Practise IB Mathematics HL 1.14 by applying complex roots and de moivre's theorem methods to questions from the Question Bank.

Exam points

Apply De Moivre’s theorem to powers and roots of complex numbers.
Determine all complex roots in polar or Euler form and verify them using complete evidence.

[Maximum number: 5]

Let $z=1-\cos 2 \theta-\mathrm{i} \sin 2 \theta, z \in \mathbb{C}, 0 \leq \theta \leq \pi$ .

Hence find the cube roots of z in modulus-argument form.

attempt to apply De Moivre's theorem
$(1-\cos 2 \theta-i \sin 2 \theta)^{\frac{1}{3}}=2^{\frac{1}{3}}(\sin \theta)^{\frac{1}{3}}\left[\cos \left(\frac{\theta-\frac{\pi}{2}+2 n \pi}{3}\right)+i \sin \left(\frac{\theta-\frac{\pi}{2}+2 n \pi}{3}\right)\right]$
Note: A1 for modulus, A1 for dividing argument of z by 3 and A1 for $2 n \pi$ .
Hence cube roots are the above expression when n=-1,0,1.
Equivalent forms are acceptable.
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IB Maths AA HL Ahl 1 14 Hl Complex Roots and De Moivres Theorem Questions question 1