Using De Moivre's theorem, express cos(3*theta) entirely in terms of powers of cos(theta).
Answer and explanation
Correct answer: C. cos(3theta) = 4cos^3(theta) - 3*cos(theta)
- Step 1: Step 1: (cos(theta) + isin(theta))^3 = cos(3theta) + isin(3theta). Binomial expansion: cos^3(theta) + 3icos^2(theta)sin(theta) - 3cos(theta)sin^2(theta) - isin^3(theta). Step 2. Step 2: Equate real parts: cos(3theta) = cos^3(theta) - 3cos(theta)sin^2(theta). Substitute sin^2(theta) = 1 - cos^2(theta): cos(3theta) = cos^3(theta) - 3cos(theta)(1 - cos^2(theta)) = cos^3(theta) - 3cos(theta) + 3cos^3(theta) = 4cos^3(theta) - 3cos(theta). Answer: cos(3theta) = 4cos^3(theta) - 3cos(theta). Why not the others:** This identity is a cornerstone of trigonometric polynomial representation.
Practise more Complex Numbers, De Moivre's Theorem and Polynomial Theory.
Free questions with an explanation for every one, and full timed New Zealand Scholarship mock tests.
More Complex Numbers, De Moivre's Theorem and Polynomial Theory questions
- What is the modulus and principal argument of the complex number z = -1 + i*sqrt(3)?
- Using De Moivre's theorem, evaluate (1 + i)^8.
- If 2 - 3i is a root of the cubic equation x^3 - 3x^2 + 9x + 13 = 0 with real coefficients, what are the other two roots?
- What geometric locus is described by the equation |z - (2 + 3i)| = 5 in the complex Argand plane?
- What geometric locus is described by |z - 4| = |z + 2i| in the complex plane?
- What are the three cube roots of unity (solutions to z^3 = 1) in exponential form?