Problem 60 - Algebra and Trigonometry

Question

In Exercises \(53-64,\) use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. $$ \left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^{6} $$

Step-by-Step Solution

Answer

The \( [\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)]^{6} \) in rectangular form is \( 13.5 - 13.5\sqrt{3}i \).

Step 1: Identify the components of the complex number

Here, the complex number is given as \( [\sqrt{3}(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18})] \). Therefore, \( r = \sqrt{3} \), \( \theta = \frac{5 \pi}{18} \) and \( n = 6 \).

Step 2: Apply DeMoivre's theorem to calculate the power of the complex number

Following DeMoivre’s theorem we calculate \( r^n (\cos n\theta + i \sin n\theta) \). Substituting for \( r \), \( \theta \) and \( n \), we get \( (\sqrt{3})^6 (\cos 6\frac{5 \pi}{18} + i \sin 6\frac{5 \pi}{18}) \). Simplifying this, we arrive at \( 27 (\cos \frac{5 \pi}{3} + i \sin \frac{5 \pi}{3}) \)

Step 3: Convert from polar to rectangular form

Using the identities \( \cos \frac{5 \pi}{3} = \frac{1}{2} \) and \( \sin \frac{5 \pi}{3} = -\frac{\sqrt{3}}{2} \). Substituting these into the equation, we obtain \( 27 (\frac{1}{2} -i \frac{\sqrt{3}}{2}) \), which simplifies to \(13.5 - 13.5\sqrt{3}i \)

Key Concepts

Complex Numbers

Complex numbers are one of the most fascinating concepts in mathematics, bridging the gap between algebra and geometry. A complex number consists of two parts: a real part and an imaginary part. The standard form of a complex number is expressed as \(a + bi\), where \(a\) represents the real part, \(b\) represents the imaginary part, and \(i\) is the square root of \(-1\). The beauty of complex numbers lies in their ability to perform operations that real numbers cannot, such as taking the square root of negative numbers.

In the original exercise, the complex number is given in polar form, which is another way to express complex numbers. It highlights a number's magnitude and direction. It's like pointing to a spot in a two-dimensional space, saying 'it's this far away (the radius \(r\)) and in that direction (the angle \(\theta\)) from the origin.' This form is incredibly useful when dealing with multiplication, division, and powers of complex numbers, as seen with DeMoivre's Theorem.

Polar to Rectangular Form

Conversion from polar to rectangular form is a critical skill when working with complex numbers. It's like translating between two languages of the mathematical world. Polar form uses the radius \(r\) and the angle \(\theta\), and is written as \(r(\text{cos}\, \theta + i\text{sin}\, \theta)\). Rectangular form, on the other hand, uses the x-axis and y-axis coordinates, akin to plotting points on a graph, and looks like \(a + bi\).

To convert from polar to rectangular form, one uses the cosine and sine of the angle for the real and imaginary parts respectively, multiplied by the radius. Essentially, you are using the angle to find out how far you move horizontally (the real part) and how far you move vertically (the imaginary part). The step-by-step solution demonstrates this by first finding the cosine and sine values of the angle and then multiplying both by the radius (here, raised to the appropriate power), to arrive at the rectangular form.

Powers of Complex Numbers

Finding the powers of complex numbers can be quite challenging if we stick to the rectangular form. DeMoivre's Theorem offers an elegant solution to this problem. It states that to raise a complex number \(z = r(\text{cos}\, \theta + i\text{sin}\, \theta)\) to a power \(n\), we can simply raise the radius \(r\) to the \(n^{\text{th}}\) power and multiply the angle \(\theta\) by \(n\). This results in the new complex number \(r^n(\text{cos}\, n\theta + i\text{sin}\, n\theta)\).

This transformation makes exponential growth or decay within the complex plane extremely manageable. The exercise solution beautifully illustrates the application of DeMoivre's Theorem by raising the radius to the sixth power and multiplying the angle by six. Upon computing we have a new radius and a new angle which, when converted back to rectangular form using cosine and sine, yields the complex number's power in a far simpler, elegant and comprehensible manner than if we attempted to multiply the number by itself repeatedly in rectangular form.