Problem 74 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[2\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\right]^{8}$$

Step-by-Step Solution

Answer

The power of the complex number is \( 256 \).

Step 1: Convert the Complex Number to Polar Form

The complex number is already given in polar form \( r(cos \theta + i sin \theta) \), where \( r=2 \), \( \theta=\frac{\pi}{2} \) and we have to compute the power of \( 8 \).

Step 2: Application of DeMoivre's Theorem

According to DeMoivre's theorem, \( [r(cos \theta + i sin \theta)]^n = r^n(\cos n\theta + i \sin n\theta) \). Here \( r=2 \), \( \theta=\frac{\pi}{2} \) and \( n=8 \), substituting these values we get \( \left[2(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})\right]^{8} = 2^8(\cos(8*\frac{\pi}{2}) + i \sin(8*\frac{\pi}{2})) \).

Step 3: Simplifying the Expression

After simplifying, it becomes \( 256(\cos 4\pi + i \sin 4\pi) \). The \(\cos 4\pi = 1\) and \(\sin 4\pi = 0\), substituting these values we get \( 256(1 + i * 0) = 256 \).

Key Concepts

Complex Numbers

Complex numbers form a number system that expands the traditional notion of numbers to include the square root of negative values. A complex number is written as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, satisfying the equation \( i^2 = -1 \). Unlike real numbers, which can be plotted on a one-dimensional number line, complex numbers require a two-dimensional plane for graphical representation called the complex plane. Here, the 'x' axis represents the real part of the numbers, while the 'y' axis corresponds to the imaginary part.

Understanding complex numbers is essential for solving equations that have no real solutions, as well as for various applications in science and engineering, such as signal analysis and quantum physics.

For example, the complex number \( 3 + 4i \) consists of a real part, 3, and an imaginary part, 4i. These components allow for rich mathematical exploration and properties distinct from the real numbers.

Polar Form of Complex Numbers

The polar form of a complex number is a different way of representing complex numbers, involving an angle and a radius, which can be particularly useful when dealing with multiplication or division of complex numbers, or finding their powers or roots. Any complex number \( a + bi \) can be represented in polar form as \( r(cos \theta + i sin \theta) \), where \( r \) is the magnitude (or modulus) of the complex number and is given by \( \sqrt{a^2 + b^2} \), and \( \theta \) is the argument, which is the angle formed with the positive direction of the x-axis in the complex plane.

In handling mathematical problems, converting to polar form greatly simplifies the computations, particularly for multiplication or raising complex numbers to a power. For instance, a complex number like \( 2(cos \frac{\theta}{2} + i sin \frac{\theta}{2}) \) clearly shows its magnitude of 2 and angle \( \frac{\theta}{2} \). This form is instrumental in understanding DeMoivre's Theorem and its applications.

Powers of Complex Numbers

Calculating the powers of complex numbers can be greatly simplified by applying DeMoivre's Theorem. This theorem states that for any complex number in polar form \( r(cos \theta + i \theta) \) and any integer \( n \), the power \( n \) of the complex number is given by:\[ [r(cos \theta + i sin \theta)]^n = r^n(cos n\theta + i sin n\theta) \].

As we can see from this formula, finding the power involves raising the magnitude, \( r \), to the power of \( n \) and multiplying the angle, \( \theta \), by \( n \). This method drastically reduces the complexity of the problem, especially when dealing with high powers.

For example, the exercise mentioned involves raising the complex number \( 2(cos \frac{\theta}{2} + i sin \frac{\theta}{2}) \) to the 8th power. By using DeMoivre's Theorem, we avoid the tedium of multiplying this complex number by itself eight times, and instead quickly arrive at the simplified result, which turns out to be a real number 256, due to the angle \( \theta \) being a multiple of \( 2\theta \). This illustrates the power and convenience of DeMoivre's Theorem in mathematical computations involving complex numbers.