Problem 76 - Precalculus with Limits

Question

In Exercises 67-82, use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. \(\left[2\left(\cos \dfrac{\pi}{2} + i\ \sin \dfrac{\pi}{2}\right)\right]^8\)

Step-by-Step Solution

Answer

The eighth power of provided complex number is \(256\).

Step 1: Convert into Polar Form

The originally provided complex number is in a trigonometric form which is kind of polar form already: \(2(\cos \dfrac{\pi}{2} + i\ \sin \dfrac{\pi}{2})\). It means that the modulus (r) of these complex numbers is 2, while argument (\(\theta\)) is \(\dfrac{\pi}{2}\).

Step 2: Apply de Moivre’s Theorem

According to de Moivre's theorem, \((r(cos\ \theta+ i\ sin\ \theta))^n = r^n (cos( n \theta) + i\ sin( n \theta))\). We can apply de Moivre's theorem to find the 8th power of the complex number. The result is: \(2^8 (\cos (\dfrac{8\pi}{2}) + i\ \sin (\dfrac{8\pi}{2}))\).

Step 3: Simplify the Expression

After simplification, we get \(256 (\cos 4\pi + i\ \sin 4\pi)\) . Now, recall that \(\cos n\pi\) will be 1 for even n and \(\sin n\pi\) will be 0 for all n. Therefore, result simplifies into: \(256 (1 + i\ 0)\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers, vital for solving equations that can't be solved using only real numbers. In simple terms, a complex number is of the form 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.

For example, in the complex number 3 + 4i, 3 is the real component and 4i is the imaginary component. Complex numbers can be added, subtracted, multiplied, and even divided by using algebraic rules combined with the fact that i^2 is -1. These operations may lead to expressions that can still be simplified by this property of i.

Polar Form of Complex Numbers

The polar form of a complex number is another way to represent complex numbers, emphasizing their magnitude and direction in the complex plane. Instead of using a + bi, the polar form uses r(cos(θ) + i sin(θ)), where r is the modulus (distance from the origin to the point in the complex plane) and θ is the argument (the angle measured from the positive real axis to the line segment connecting the origin to the point).

This representation is particularly useful when dealing with multiplication or finding powers of complex numbers because it simplifies the process by using angle addition or multiplication. For instance, the modulus of the complex number will be raised to the power, and the argument will be multiplied by the power.

Trigonometric Form of Complex Numbers

The trigonometric form of complex numbers is closely related to the polar form, expressed as r(cos(θ) + i sin(θ)). It is so named because it uses trigonometric functions to describe the location of the point. The main benefit of this form is when working with complex number powers and roots, as it simplifies the calculations.

The relationship between an angle in a unit circle and the sine and cosine functions is at the core of its usefulness. The trigonometric form makes operations like multiplying, dividing, and raising to powers more intuitive, especially when considering the geometric interpretations of these operations.

Powers of Complex Numbers

Calculating the powers of complex numbers may seem daunting, but it becomes much easier when using DeMoivre's Theorem, especially when the complex number is expressed in trigonometric form. DeMoivre's Theorem states that (r(cos(θ) + i sin(θ)))^n = r^n (cos(nθ) + i sin(nθ)).

This elegant theorem simplifies the computing of powers to just raising the modulus to the power and multiplying the angle by the power. In practice, for example, to find the 8th power of 2(cos(π/2) + i sin(π/2)), one would calculate 2^8 and multiply π/2 by 8, resulting in a much simpler final expression. Remember to normalize the resulting angle, since trigonometric functions are periodic.