De Moivre’s theorem (HSC SSCE Mathematics Extension 2): Flashcards
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What does De Moivre's Theorem state?
What does De Moivre's Theorem state?
For integer n, z^n = r^n (cos nθ + i sin nθ).
What is a complex number?
What is a complex number?
A number in the form a + bi, with real and imaginary parts.
How is the modulus of a complex number defined?
How is the modulus of a complex number defined?
|z| = √(a² + b²), showing distance from origin.
What is the polar form of a complex number?
What is the polar form of a complex number?
z = r(cos θ + i sin θ), with r as modulus and θ as angle.
What is the argument of a complex number?
What is the argument of a complex number?
The angle θ with the positive real axis on the Argand diagram.
What is (cos π/3 + i sin π/3)³ using De Moivre's theorem?
What is (cos π/3 + i sin π/3)³ using De Moivre's theorem?
Result is 1 + 0i after simplification.
What is the base case for proving De Moivre's Theorem?
What is the base case for proving De Moivre's Theorem?
Check the theorem for n = 1.
How do you find the 4th roots of 16(cos π + i sin π)?
How do you find the 4th roots of 16(cos π + i sin π)?
Use z^(1/4) = r^(1/4)(cos(θ + 2πk)/4 + i sin(θ + 2πk)/4).
What is a trigonometric identity from De Moivre's Theorem?
What is a trigonometric identity from De Moivre's Theorem?
Expand (cos x + i sin x)² to relate to known identities.
What should you remember about De Moivre's Theorem?
What should you remember about De Moivre's Theorem?
Be cautious with non-integer powers; it applies to integers.