Problem 13 - Advanced Engineering Mathematics

Question

Write the number given in polar form in the form \(a+i b\). $$ z=6\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) $$

Step-by-Step Solution

Answer

\(z = 3 \sqrt{2+\sqrt{2}} + 3i \sqrt{2-\sqrt{2}}\)

Step 1: Identify the Polar Form

The number given is presented in polar form: \(z = 6 (\cos \frac{\pi}{8} + i \sin \frac{\pi}{8})\), where 6 is the modulus and \(\frac{\pi}{8}\) is the argument of the complex number.

Step 2: Use Euler's Formula

Recall that \(\cos \theta + i \sin \theta = e^{i\theta}\). Thus, \(z = 6 e^{i \frac{\pi}{8}}\). However, since we need the rectangular form, we'll focus on calculating \(\cos \frac{\pi}{8}\) and \(\sin \frac{\pi}{8}\).

Step 3: Calculate \(\cos \frac{\pi}{8}\) and \(\sin \frac{\pi}{8}\)

Using the half-angle formulas, we find \(\cos \frac{\pi}{8} = \sqrt{\frac{1 + \cos \frac{\pi}{4}}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2+\sqrt{2}}{4}}\) and \(\sin \frac{\pi}{8} = \sqrt{\frac{1 - \cos \frac{\pi}{4}}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \sqrt{\frac{2-\sqrt{2}}{4}}\).

Step 4: Substitute the Values into Polar Form

Now that we have \(\cos \frac{\pi}{8}\) and \(\sin \frac{\pi}{8}\), plug them into the original polar expression: \(z = 6 \left(\sqrt{\frac{2+\sqrt{2}}{4}} + i \sqrt{\frac{2-\sqrt{2}}{4}}\right)\).

Step 5: Simplify the Expression

Distribute the 6: \(z = 6 \sqrt{\frac{2+\sqrt{2}}{4}} + 6i \sqrt{\frac{2-\sqrt{2}}{4}}\). This simplifies to \(z = 6 \cdot \frac{\sqrt{2+\sqrt{2}}}{2} + 6i \cdot \frac{\sqrt{2-\sqrt{2}}}{2}\), or \(z = 3 \sqrt{2+\sqrt{2}} + 3i \sqrt{2-\sqrt{2}}\).

Step 6: Verify Rectangular Form

The expression now is written in the form \(a + ib\), which confirms the polar conversion. Thus, \(z = 3 \sqrt{2+\sqrt{2}} + 3i \sqrt{2-\sqrt{2}}\).

Key Concepts

Polar to Rectangular Conversion

Complex numbers are often used in the polar coordinate system. They are typically represented in the form \( z = r(\cos \theta + i \sin \theta) \), where:

• \( r \) is the modulus, representing the distance from the origin.
• \( \theta \) is the argument, representing the angle from the positive x-axis.

To convert a complex number from polar to rectangular form, use the relationships:

• Real part: \( x = r \cos \theta \)
• Imaginary part: \( y = r \sin \theta \)

By calculating the cosine and sine of the angle \( \theta \), the complex number is expressed as \( x + iy \). This form shows how far along the real and imaginary axes the number stretches, providing an easier visualization on the complex plane.

Euler's Formula

Euler's formula is a beautiful bridge between exponential functions and trigonometry. It states that for any real number \( \theta \),\[ e^{i\theta} = \cos \theta + i \sin \theta \]This formula is extremely useful for converting complex numbers between different forms, particularly in polar coordinates. It shows that complex exponentials can be expressed in terms of trigonometric functions.Let's see how this applies to our task:

• The complex number in polar form, \( z = 6 \left( \cos \frac{\pi}{8} + i \sin \frac{\pi}{8} \right) \), can be represented as \( 6e^{i\frac{\pi}{8}} \) using Euler's formula.
• This insight simplifies many calculations and provides a clear connection between trigonometry and complex analysis.

Understanding Euler's formula gives insight into the power and elegance of expressing rotations and waves in complex forms.

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles, particularly those concerning right triangles. They are crucial in simplifying expressions involving trigonometric functions.In our conversion task, the half-angle identities are essential. They are derived from the cosine and sine double-angle identities:

• \( \cos \frac{\theta}{2} = \sqrt{\frac{1 + \cos \theta}{2}} \)
• \( \sin \frac{\theta}{2} = \sqrt{\frac{1 - \cos \theta}{2}} \)

To find \( \cos \frac{\pi}{8} \) and \( \sin \frac{\pi}{8} \), we used these identities with \( \theta = \frac{\pi}{4} \):

• \( \cos \frac{\pi}{8} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \)
• \( \sin \frac{\pi}{8} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \)

These identities are essential tools for breaking down angles into easily computable parts, allowing complex numbers to be worked with seamlessly in both algebraic and trigonometric contexts.