Using De Moivre\'s theorem, z^4 = (cos(θ) + i sin(θ))^4 = cos(4θ) + i sin(4θ).
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?
cos(4θ) + i sin(4θ)
cos(2θ) + i sin(2θ)
cos(3θ) + i sin(3θ)
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).
De Moivre's Theorem – A formula that connects complex numbers in polar form to trigonometric functions, allowing for the computation of powers and roots of complex numbers.
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