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If z = cos(θ) + i sin(θ), what is z^4?

Practice Questions

Q1
If z = cos(θ) + i sin(θ), what is z^4?
  1. cos(4θ) + i sin(4θ)
  2. cos(2θ) + i sin(2θ)
  3. cos(3θ) + i sin(3θ)
  4. cos(θ) + i sin(θ)

Questions & Step-by-Step Solutions

If z = cos(θ) + i sin(θ), what is z^4?
  • Step 1: Start with the given expression z = cos(θ) + i sin(θ).
  • Step 2: Recognize that we need to find z^4, which means we will raise z to the power of 4.
  • Step 3: Use De Moivre's theorem, which states that (cos(θ) + i sin(θ))^n = cos(nθ) + i sin(nθ).
  • Step 4: In our case, n = 4, so we apply the theorem: z^4 = (cos(θ) + i sin(θ))^4.
  • Step 5: According to De Moivre's theorem, this simplifies to z^4 = cos(4θ) + i sin(4θ).
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