Question Detail

MTH 101 MTH 101 Past Questions Objective Question
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Question

If \( z = 4(\cos 110^\circ + i \sin 110^\circ) \), find \( z^3 \).
Options
A \(16(\cos 330^\circ + i \sin 330^\circ)\)
B \(4(\cos 330^\circ + i \sin 330^\circ)\)
C \(64(\cos 330^\circ - i \sin 330^\circ)\)
D \(64(\cos 330^\circ + i \sin 330^\circ)\)
Correct Answer
Correct Answer

Option D is the correct answer.

Detailed Explanation

By De Moivre's Theorem: z^n = r^n (cos(nθ) + i sin(nθ)). Here, r = 4, θ = 110°, n = 3. So z^3 = 4^3 (cos(3110°) + i sin(3110°)) = 64 (cos(330°) + i sin(330°)).

Hint

Apply De Moivre's Theorem: (r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ)).

Question Info

Course: MTH 101
Topic: MTH 101 Past Questions
Type: Objective
Difficulty: Not specified
Question 507 of 696
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