De Moivre's Theorem

Question

Using De Moivre's theorem, the value of the expression (cos 15 degrees + i sin 15 degrees)^6 simplifies to which complex number in rectangular form?

RankGuru · Accepted Answer

De Moivre's theorem states that (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ), the workhorse identity for powers of complex numbers in JEE Advanced. Here θ = 15 degrees and n = 6, so nθ = 6·15 = 90 degrees and the expression equals cos 90 degrees + i sin 90 degrees = 0 + i·1 = i. Option 1/2 + i√3/2 corresponds to 60 degrees, an arithmetic slip in nθ. Option √3/2 + i/2 corresponds to 30 degrees. Option -1/2 + i√3/2 corresponds to 120 degrees. Hence the correct value is i. Plausibility check: the modulus of the base is 1, so any power also has modulus 1, and i indeed lies on the unit circle at angle 90 degrees, matching 6 times 15 degrees exactly. De Moivre's theorem is the gateway to extracting roots of complex numbers and to deriving multiple-angle trigonometric identities, both of which recur throughout the JEE Advanced syllabus. Because the base here has unit modulus, every power also has unit modulus and merely rotates around the circle, so tracking the accumulated angle is the entire computation and any magnitude change would signal an arithmetic mistake.

De Moivre's Theorem

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