Problem 14 - Advanced Engineering Mathematics

Question

Write the number given in polar form in the form \(a+i b\). $$ z=10\left(\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \( 8.090 + i 5.878 \).

Step 1: Identify the given polar form

The given complex number is in polar form: \( z = 10 \left( \cos \frac{\pi}{5} + i \sin \frac{\pi}{5} \right) \). This form can be expressed as \( z = r(\cos \theta + i \sin \theta) \) where \( r = 10 \) is the modulus and \( \theta = \frac{\pi}{5} \) is the argument.

Step 2: Convert polar to rectangular form

To convert from polar to rectangular form, use the formulas: \( a = r \cos \theta \) and \( b = r \sin \theta \). Meticulously compute each part:

Step 3: Calculate the real part

For the real part, use \( a = 10 \cos \frac{\pi}{5} \). Calculate \( \cos \frac{\pi}{5} \): \[ \cos \frac{\pi}{5} = \cos 36^\circ \approx 0.8090 \] Thus, \[ a = 10 \times 0.8090 = 8.090 \]

Step 4: Calculate the imaginary part

For the imaginary part, use \( b = 10 \sin \frac{\pi}{5} \). Calculate \( \sin \frac{\pi}{5} \): \[ \sin \frac{\pi}{5} = \sin 36^\circ \approx 0.5878 \]Thus, \[ b = 10 \times 0.5878 = 5.878 \]

Step 5: Formulate the rectangular form

Combine the real and imaginary parts to express \( z \) in rectangular form: \[ z = a + ib = 8.090 + i 5.878 \]

Key Concepts

Polar Form

Complex numbers can be represented in different ways, and one such way is the polar form. The polar form of a complex number expresses it in terms of a distance (modulus) and an angle (argument). This can be very intuitive when you think of a number as a point in the two-dimensional plane. Here, the modulus represents the distance from the origin to the point, and the argument represents the angle from the positive x-axis to the line connecting the origin to the point.

The polar form is typically described as:

• The modulus \( r \) is the length of the vector from the origin to the point.
• The argument \( \theta \) is the angle between the positive x-axis and the line connecting the origin with the point.

In mathematical terms, the polar form of a complex number \( z \) is:\[ z = r (\cos \theta + i \sin \theta) \]Here, \( r \) is the modulus, and \( \theta \) is the argument.
This form is particularly useful in problems involving multiplication and division of complex numbers, as it simplifies many operations. If you have a complex number in polar form such as \( z = 10 (\cos \frac{\pi}{5} + i \sin \frac{\pi}{5} ) \), \( r = 10 \) and \( \theta = \frac{\pi}{5} \). This provides a clear geometric interpretation and simplifies trigonometric calculations.

Rectangular Form

The rectangular form of a complex number is the standard and most commonly used representation, where it is expressed in terms of its real and imaginary components. It often helps in understanding calculations, as it shows the complex number as a simple combination of real and imaginary parts.

In rectangular form, a complex number \( z \) is written as:\[ z = a + ib \]where:

• \(a\) is the real part of the complex number.
• \(b\) is the imaginary part of the complex number.

These components are directly linked to the Cartesian coordinates of the complex number in the plane. For a complex number originally given in polar form, you can convert it to rectangular form using:

• \( a = r \cos \theta \)
• \( b = r \sin \theta \)

This conversion allows you to easily add, subtract, or compare complex numbers. Using the example \( z = 10 (\cos \frac{\pi}{5} + i \sin \frac{\pi}{5} ) \), we find that the real part \( a = 10 \times 0.8090 = 8.090 \) and the imaginary part \( b = 10 \times 0.5878 = 5.878 \). Thus, the rectangular form of the given complex number is \( z = 8.090 + i5.878 \).

Trigonometry

Trigonometry plays a crucial role in converting between the polar and rectangular forms of complex numbers. It essentially helps us deal with the relationships between angles and distances. When a complex number is written in polar form, trigonometric functions like cosine and sine become central to expressing the components of the complex number in the plane.

To convert from polar to rectangular coordinates, we leverage the following trigonometric identities:

• \( \cos \theta \) represents how far along the x-axis (real part) the point is.
• \( \sin \theta \) represents how far along the y-axis (imaginary part) the point is.

Consider the equation \( z = 10 (\cos \frac{\pi}{5} + i \sin \frac{\pi}{5} ) \). Here:

• \( \cos \frac{\pi}{5} \approx 0.8090 \)
• \( \sin \frac{\pi}{5} \approx 0.5878 \)

These calculations allow you to break down the complex number into its rectangular components by multiplying the modulus by these trigonometric values. Understanding these basics of trigonometry helps simplify and solve complex number problems by making the transition between different forms more intuitive.