Problem 9 - Advanced Engineering Mathematics

Question

$$\frac{3 \sqrt{2}}{2}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)$$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \(-\frac{3}{2} - \frac{3}{2}i\).

Step 1: Understanding the Format

The exercise gives a complex number in trigonometric (polar) form: \[ \frac{3 \sqrt{2}}{2}\left(\cos \frac{5 \pi}{4} + i \sin \frac{5 \pi}{4}\right) \].This is a polar representation where \( r = \frac{3 \sqrt{2}}{2} \) is the magnitude and \( \theta = \frac{5 \pi}{4} \) is the angle.

Step 2: Converting to Rectangular Form

To convert from polar to rectangular form, use the identities:\[ z = r (\cos \theta + i \sin \theta) = r e^{i\theta} \]. Thus, compute:\[ \frac{3 \sqrt{2}}{2} \cos \frac{5 \pi}{4} = \frac{3 \sqrt{2}}{2} \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3}{2} \].And for the imaginary part:\[ \frac{3 \sqrt{2}}{2} \sin \frac{5 \pi}{4} = \frac{3 \sqrt{2}}{2} \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3}{2}i \].

Step 3: Final Calculation

Combine the real and the imaginary parts to get the complex number in rectangular form:\[ z = -\frac{3}{2} - \frac{3}{2}i \].

Key Concepts

Polar Form

Polar form is a way of representing complex numbers using a combination of their magnitude and angle. This form is very useful in certain contexts, like when multiplying or dividing complex numbers. The general expression for a complex number in polar form is \( r(\cos \theta + i \sin \theta) \), where:

• \( r \) is the magnitude (or modulus) of the complex number, the distance from the origin in the complex plane.
• \( \theta \) is the angle (or argument), measured in radians from the positive real axis.

A key benefit of the polar form is how it simplifies the operation of complex numbers, especially in calculations involving powers and roots. This is because the polar representation aligns well with Euler's formula, \( e^{i\theta} = \cos \theta + i \sin \theta \), allowing for the expression of complex numbers as \( r e^{i\theta} \).
In our exercise, \( r = \frac{3 \sqrt{2}}{2} \) and \( \theta = \frac{5 \pi}{4} \), offering a succinct way to understand the complex number's properties.

Rectangular Form

Rectangular form is a more intuitive way of representing complex numbers, especially for those familiar with the Cartesian coordinate system. In the rectangular form, a complex number is expressed as \( z = a + bi \), where:

• \( a \) is the real part, corresponding to the distance along the real axis.
• \( b \) is the imaginary part, corresponding to the distance along the imaginary axis.

This form is particularly practical for addition and subtraction of complex numbers. Converting from polar to rectangular form involves calculating \( a \) and \( b \) using the equations \( a = r \cos \theta \) and \( b = r \sin \theta \).
In our solution, we found the rectangular form of the complex number as \( z = -\frac{3}{2} - \frac{3}{2}i \), where both real and imaginary parts are derived by multiplying the magnitude \( r \) with the cosine and sine of the angle \( \theta \) respectively.

Trigonometric Form

The trigonometric form, also known as polar form, is a crucial way to express complex numbers, blending the concepts of angle and magnitude. This form emphasizes how complex numbers relate to the unit circle and is given by \( z = r(\cos \theta + i \sin \theta) \).

• This form is ideal for understanding complex numbers geometrically since it correlates directly with their position on the complex plane.
• It introduces the significance of angles combined with the radial distance to define the complex number.
• It's especially advantageous for operations like multiplication and division, where the magnitudes multiply and angles add, simplifying calculations.

In our exercise, the trigonometric representation \( \frac{3 \sqrt{2}}{2}(\cos \frac{5 \pi}{4} + i \sin \frac{5 \pi}{4}) \) offers insight into both the direction and size of the complex number, paving a seamless path to converting it to its rectangular form.