IB Maths AA HL 1.14 Complex roots and De Moivre's theorem Question Bank

Practise IB Mathematics HL 1.14 by applying complex roots and de moivre's theorem methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[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

M1
$(1-\cos 2 \theta-i \sin 2 \theta)^{\frac{1}{3}}=2^{\frac{1}{3}}(\sin \theta)^{\frac{1}{3}}\left[\cos \left(\frac{\theta-\frac{\pi}{2}+2 n \pi}{3}\right)+i \sin \left(\frac{\theta-\frac{\pi}{2}+2 n \pi}{3}\right)\right]$
A1A1A1
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.
A1
Total

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AHL 1.14 (HL)—Complex roots and De Moivre's theorem question 1