De Moivre's Theorem & Complex Numbers

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Questions and Answers

Given $z = 2(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))$, what is $z^6$?

-64 (correct)
64i
-64i
64

If a polynomial equation with real coefficients has a complex root $3 + 4i$, then $3 - 4i$ must also be a root.

True (A)

What is the sum of the 5th roots of unity?

De Moivre's Theorem states that for any complex number in polar form $z = r(\cos \theta + i \sin \theta)$ and any integer n, $z^n = r^n(\cos n\theta + i \sin n\theta)$. Therefore, $(\cos \theta + i \sin \theta)^n = $ ______

cos nθ + i sin nθ Signup and view all the answers

Match each complex number operation with its corresponding geometric transformation in the complex plane:

Multiplication by $i$ = Rotation by 90 degrees counterclockwise Multiplication by a real number $r > 1$ = Dilation (scaling away from the origin) Addition of a complex number $a + bi$ = Translation Taking the complex conjugate = Reflection about the real axis Signup and view all the answers

Which of the following statements regarding the nth roots of unity is NOT true?

They all lie on the real axis. (D) Signup and view all the answers

The modulus of a complex number $z = a + bi$ is given by $\sqrt{a^2 - b^2}$.

False (B) Signup and view all the answers

Express the complex number $z = 1 + i\sqrt{3}$ in polar form $re^{i\theta}$. Give exact values.

2e^(iπ/3) Signup and view all the answers

According to the Fundamental Theorem of Algebra, every non-constant single-variable polynomial with complex coefficients has at least ______ complex root.

one Signup and view all the answers

Using De Moivre's Theorem, derive an expression for $\cos(3\theta)$ in terms of $\cos(\theta)$. Which of the following is correct?

$4\cos^3(\theta) - 3\cos(\theta)$ (B) Signup and view all the answers

Flashcards

Complex Number

A number in the form a + bi, where a and b are real numbers, and i is the imaginary unit (i² = -1).

Polar Form of a Complex Number

Representation of a complex number z = a + bi as z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.

De Moivre's Theorem

For z = r(cos θ + i sin θ), zⁿ = rⁿ(cos nθ + i sin nθ).

nth Roots of Unity

Solutions to zⁿ = 1, given by z_k = cos(2πk/n) + i sin(2πk/n) for k = 0, 1, 2,..., n-1.

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Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one complex root.

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Complex Conjugate Root Theorem

If a + bi is a root of a polynomial with real coefficients, then a - bi is also a root.

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De Moivre's Theorem and Trigonometry

Using De Moivre's Theorem to derive formulas like sin(nθ) and cos(nθ) in terms of sin(θ) and cos(θ).

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Study Notes

De Moivre's Theorem provides a formula for calculating powers and roots of complex numbers.

Complex Numbers

A complex number is expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property i² = -1.
The real part of a complex number z is denoted as Re(z) = a, and the imaginary part is denoted as Im(z) = b.
Complex numbers can be represented on an Argand diagram, where the x-axis represents the real part and the y-axis represents the imaginary part.

Polar Form

A complex number z = a + bi can be represented in polar form as z = r(cos θ + i sin θ), where r is the modulus (or absolute value) of z, and θ is the argument of z.
The modulus r is calculated as r = √(a² + b²).
The argument θ is the angle between the positive real axis and the line connecting the origin to the point representing z on the Argand diagram; θ can be found using θ = tan⁻¹(b/a), adjusting for the correct quadrant.
The polar form is also written as z = re^(iθ) using Euler's formula, where e^(iθ) = cos θ + i sin θ.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form z = r(cos θ + i sin θ) and any integer n, the following holds: [r(cos θ + i sin θ)]^n = r^n(cos nθ + i sin nθ) or (cos θ + i sin θ)^n = cos nθ + i sin nθ.
This theorem is used to find powers of complex numbers.
Also, it is used to derive trigonometric identities.
For example, expanding (cos θ + i sin θ)³ using the binomial theorem and then applying De Moivre's Theorem yields expressions for cos 3θ and sin 3θ in terms of cos θ and sin θ.

nth Roots of Unity

The nth roots of unity are the solutions to the equation z^n = 1, where z is a complex number.
In polar form, the nth roots of unity are given by z_k = cos(2πk/n) + i sin(2πk/n) for k = 0, 1, 2, ..., n-1.
These roots are equally spaced around the unit circle in the complex plane.
The sum of the nth roots of unity is always zero.
The nth roots of unity form the vertices of a regular n-gon inscribed in the unit circle.

Roots of Polynomial Equations

The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
A polynomial of degree n has exactly n complex roots, counting multiplicities.
If a polynomial has real coefficients, any non-real roots occur in conjugate pairs. That is, if a + bi is a root, then a - bi is also a root.
Roots of polynomial equations can be found using various methods, including factoring, synthetic division, and numerical methods.
Complex roots of polynomial equations can be expressed in polar form.

Applications in Trigonometry

De Moivre's Theorem can be used to derive multiple angle formulas in trigonometry.
For example, by using De Moivre's Theorem, trigonometric identities for sin(nθ) and cos(nθ) can be derived in terms of sin(θ) and cos(θ).
It can be used to find trigonometric values of angles.
It simplifies complex trigonometric expressions.
The representation of complex numbers in polar form simplifies many trigonometric problems.