Question

Use De Moivre's theorem to show that if \(z=\cos \theta+\mathrm{j} \sin \theta\) then (a) \(z^{n}=\cos n \theta+j \sin n \theta\) (b) \(z^{-n}=\cos n \theta-\mathrm{j} \sin n \theta\) Deduce that $$ z^{n}+\frac{1}{z^{n}}=2 \cos n \theta $$ and $$ z^{n}-\frac{1}{z^{n}}=2 \mathrm{j} \sin n \theta $$

Short answer

Question: Use De Moivre's theorem to deduce the following expressions for a complex number $z = \cos\theta + \mathrm{j}\sin\theta$, where $n$ is an integer: (a) $z^n + \frac{1}{z^n} = 2\cos n\theta$ (b) $z^n - \frac{1}{z^n} = 2\mathrm{j}\sin n\theta$ Answer: Using De Moivre's theorem, we can derive the expressions $z^n = \cos n\theta + \mathrm{j}\sin n\theta$ and $z^{-n} = \cos n\theta - \mathrm{j}\sin n\theta$. Adding these two expressions, we get $z^n + \frac{1}{z^n} = 2\cos n\theta$. Subtracting the second expression from the first one, we get $z^n - \frac{1}{z^n} = 2\mathrm{j}\sin n\theta$. Thus, we have deduced the given expressions.

Step-by-step solution

Use De Moivre's theorem to find \(z^n\) and \(z^{-n}\)

Applying De Moivre's theorem to \(z = \cos\theta + \mathrm{j}\sin\theta\), we get (a) \(z^n = \cos n\theta + \mathrm{j}\sin n\theta\) (b) To find \(z^{-n}\), first recall that the reciprocal of a complex number in polar form is given by \(1/z = \cos(-\theta) + \mathrm{j}\sin(-\theta)\). Therefore, we can find \(z^{-n}\) by taking \(z = \cos(-\theta) + \mathrm{j}\sin(-\theta)\) and applying De Moivre's theorem to get \(z^{-n} = \cos n(-\theta) + \mathrm{j}\sin n(-\theta) = \cos n\theta - \mathrm{j}\sin n\theta\)

Deduce the given expressions involving \(\cos n\theta\) and \(\sin n\theta\)

Now that we have expressions for \(z^n\) and \(z^{-n}\), we can deduce the given expressions: (a) Adding the expressions for \(z^n\) and \(z^{-n}\), we have: $$ z^{n}+\frac{1}{z^{n}}=(\cos n\theta + \mathrm{j}\sin n \theta)+(\cos n\theta -\mathrm{j}\sin n\theta)=2\cos n\theta $$ (b) Subtracting the expression for \(z^{-n}\) from the expression for \(z^n\), we get: $$ z^{n}-\frac{1}{z^{n}}=(\cos n\theta + \mathrm{j}\sin n\theta)-(\cos n\theta - \mathrm{j}\sin n\theta)=2\mathrm{j}\sin n\theta $$ Thus, we have deduced the given expressions: $$ z^{n}+\frac{1}{z^{n}}=2 \cos n \theta $$ and $$ z^{n}-\frac{1}{z^{n}}=2 \mathrm{j} \sin n \theta $$

Key concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are fundamental in advanced mathematics and engineering. They consist of two parts: a real part and an imaginary part. A complex number is generally represented in the form of a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit with the property that i^2 = -1.

Complex numbers can be added, subtracted, multiplied, and divided like real numbers, with additional rules for the imaginary unit. They are particularly useful in solving equations that have no real solutions, such as x^2 + 1 = 0.

We visualize complex numbers on a plane known as the complex plane or Argand plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This visual representation aids in better understanding operations involving complex numbers and sets the stage for the next important concept, the polar form of complex numbers.

Polar Form of Complex Numbers

The polar form of a complex number is a different way to represent complex numbers, emphasizing their magnitude and angle instead of their real and imaginary parts. In polar form, a complex number is written as r(cos θ + i sin θ), where r is the magnitude (or modulus) of the complex number and θ is the argument or angle which the line representing the complex number makes with the positive real axis in the complex plane.

The magnitude r is calculated as the square root of the sum of the squares of the real and imaginary parts, and the angle θ is determined by the trigonometric arctangent of the imaginary part divided by the real part. Converting between rectangular form (a + bi) and polar form (r(cos θ + i sin θ)) can be vital for simplifying complex number operations, especially multiplication and division.

The polar form of complex numbers is particularly central to understanding De Moivre's Theorem. This theorem, which provides a formula for raising a complex number to a power, is simpler to express and calculate using the polar form. The elegant symmetry of the polar form also helps in understanding geometric interpretations of complex number operations.

Trigonometric Identities in Complex Analysis

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides of the equation are defined. They play a crucial role in complex analysis, particularly when working with the polar form of complex numbers. These identities allow us to simplify expressions and solve equations involving complex numbers more easily.

Key trigonometric identities include the Pythagorean identities, sum and difference formulas, double angle formulas, and others. For example, the well-known Pythagorean identity sin^2 θ + cos^2 θ = 1 is crucial when normalizing the magnitude of a complex number to one, and the angle sum and difference identities are essential when multiplying or dividing two complex numbers.

These identities are integral to De Moivre's Theorem because they allow for the concise expression of powers of complex numbers and for transforming complex expressions into their trigonometric counterparts. Without these identities, complex analysis would be cumbersome and much less intuitive. Understanding and using trigonometric identities can greatly enhance one's ability to work with complex numbers and tackle advanced mathematics with confidence.