Problem 35 - Advanced Engineering Mathematics

Question

In Problems 35 and 36, express the given complex number first in polar form and then in the form \(a+i b\). $$ \left(\cos \frac{\pi}{9}+i \sin \frac{\pi}{9}\right)^{12}\left[2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\right]^{5} $$

Step-by-Step Solution

Answer

The polar form is \(32\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and the rectangular form is \(16\sqrt{3} + 16i\).

Step 1: Recognize Format of Given Complex Number

The given complex numbers are in the form \(\cos\theta + i\sin\theta\), which are written as **Euler's formula**: \(e^{i\theta}\). So we transform them into exponential form first.

Step 2: Convert to Exponential Form

Convert each term in the product to exponential form using Euler's formula.\[(\cos \frac{\pi}{9} + i \sin \frac{\pi}{9})^{12} = \left(e^{i \frac{\pi}{9}}\right)^{12} = e^{i \frac{12\pi}{9}} = e^{i \frac{4\pi}{3}},\] for the first term. \[ \left[2 \left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\right]^{5} = 2^5 \left(e^{i \frac{\pi}{6}}\right)^{5} = 32 e^{i \frac{5\pi}{6}}.\] for the second term.

Step 3: Multiply in Exponential Form

Multiply the two complex numbers in exponential form:\[ e^{i \frac{4\pi}{3}} \times 32 e^{i \frac{5\pi}{6}} = 32 e^{i (\frac{4\pi}{3} + \frac{5\pi}{6})} = 32 e^{i \frac{13\pi}{6}}\] Simplify the angle: \[ \frac{13\pi}{6} = \frac{13\pi}{6} - 2\pi = \frac{\pi}{6} \] So, the expression becomes \[ 32 e^{i \frac{\pi}{6}} \]

Step 4: Convert Back to Complex Rectangular Form

Use Euler’s formula to convert back to rectangular form:\[ e^{i \frac{\pi}{6}} = \cos \frac{\pi}{6} + i \sin \frac{\pi}{6}, \] which is \[ \frac{\sqrt{3}}{2} + i \frac{1}{2}. \] Multiply by 32: \[32 \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = 16\sqrt{3} + 16i.\]

Step 5: Final Answer

The complex number in polar form is \(32\left(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6}\right)\) and in rectangular form is \(16\sqrt{3} + 16i\).

Key Concepts

Euler's Formula

Euler’s Formula is a powerful tool for working with complex numbers, especially when they are expressed in trigonometric form. It states that for any real number \(\theta\), the expression \(e^{i\theta} = \cos\theta + i\sin\theta\). This elegant formula links complex exponentials to trigonometric functions, and it can be used to simplify complex calculations.

For the given exercise, we recognized expressions like \(\cos\frac{\pi}{9} + i \sin\frac{\pi}{9}\) as being in the trigonometric form and then used Euler's formula to convert them to the exponential form. For example:

• \((\cos \frac{\pi}{9} + i \sin \frac{\pi}{9})^{12}\) becomes \(\left(e^{i \frac{\pi}{9}}\right)^{12} = e^{i \frac{12\pi}{9}} = e^{i \frac{4\pi}{3}}\).
• Using Euler’s formula simplifies multiplication and division of complex numbers, as it allows us to handle their phase or angle component and magnitude more easily.

When working with trigonometric expressions, remember that Euler's formula is your ally, transforming them into an exponential form that's often simpler to manipulate.

Polar Form

Polar form is an alternative way to represent complex numbers. Instead of using the rectangular form \(a + ib\), complex numbers can be written in the form \(r(\cos\theta + i\sin\theta)\), where:

• \(r\) is the magnitude (or modulus) of the complex number.
• \(\theta\) is the angle (or argument) made with the positive x-axis.

In this problem, the complex numbers are already provided in a variant of polar form through their trigonometric representation. Utilizing polar form can simplify operations such as multiplication and division because it leverages the magnitudes and angles directly.

For the given term \(\left[2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})\right]^{5}\), this can be thought of as a complex number with magnitude \(2\) and angle \(\frac{\pi}{6}\). Raising to the power of 5, simply:

• The magnitude is raised to the fifth power: \(2^5 = 32\).
• The angle is multiplied by 5: \(\frac{\pi}{6} \times 5 = \frac{5\pi}{6}\).

Polar form thus allows us to utilize the magnitudes (lengths) and angles, providing a natural extension for computation in the complex plane.

Rectangular Form

Rectangular form represents complex numbers in the form \(a + ib\), where \(a\) is the real part and \(ib\) is the imaginary part. After calculations in polar or exponential form, it's often useful to convert the result back to rectangular form to better interpret and visualize it.

Using the final expression from our exercise, \(32 e^{i \frac{\pi}{6}}\), we applied Euler’s formula to revert it to rectangular coordinates. Thus:

• \(\cos \frac{\pi}{6}= \frac{\sqrt{3}}{2}\), and \(\sin \frac{\pi}{6} = \frac{1}{2}\).
• Inserting into the formula gives \(e^{i \frac{\pi}{6}} = \frac{\sqrt{3}}{2} + i \frac{1}{2}\).
• Multiplying by 32 gives \(32\left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = 16\sqrt{3} + 16i\).

This result \((16\sqrt{3} + 16i)\) is typically how complex numbers are expressed in standard math or computer applications. The rectangular form provides a straightforward understanding of the real and imaginary components of the complex number.