Problem 14 - Advanced Engineering Mathematics

Question

In Problems 11-14, 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 number in rectangular form is \( 8.09 + i 5.88 \).

Step 1: Understand the Polar Form

The given expression is in polar form, which is written as \( z = r(\cos \theta + i \sin \theta) \). Here, \( r = 10 \) and \( \theta = \frac{\pi}{5} \).

Step 2: Rewrite Using Euler's Formula

We can use Euler's formula to rewrite the polar form as \( z = re^{i\theta} \). Given, \( z = 10 \left(\cos \frac{\pi}{5} + i \sin \frac{\pi}{5}\right) \) can be expressed as \( z = 10e^{i\frac{\pi}{5}} \).

Step 3: Convert to Rectangular Form

To convert to rectangular form \( a + ib \), calculate the cosine and sine values: \( \cos \frac{\pi}{5} \approx 0.809 \) and \( \sin \frac{\pi}{5} \approx 0.588 \).

Step 4: Calculate Rectangular Coordinates

Substitute the values into the form \( z = r(\cos \theta + i \sin \theta) \). Thus, \( z = 10(0.809 + i 0.588) \).

Step 5: Simplify

Multiply both parts by 10: \( a + ib = 10 \times 0.809 + 10 \times i \times 0.588 \), simplifying to \( z = 8.09 + i 5.88 \).

Key Concepts

polar form

Complex numbers can be represented in different ways, and one of the widely used forms is the polar form. In this form, a complex number is expressed as \( z = r(\cos \theta + i \sin \theta) \). Here, \( r \) is the modulus of the complex number, representing its magnitude or distance from the origin in the complex plane. The angle \( \theta \) is the argument, which is the direction or angle the line makes with the positive x-axis.

To convert from one form to another, understanding the polar form is crucial. It provides a geometric meaning - thinking about the complex number as a point on a circle of radius \( r \). Polar form is particularly useful for multiplying and dividing complex numbers, since it simplifies the operations by rotating and scaling rather than directly calculating with components.

rectangular form

The rectangular form of a complex number, sometimes referred to as the Cartesian form, is perhaps the most straightforward way of representing complex numbers. It is expressed as \( a + ib \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number.

This form corresponds directly to coordinates on the complex plane, making it intuitive for understanding addition and subtraction of complex numbers. When converting from polar to rectangular form, the real part \( a \) can be found using \( a = r \cos \theta \), and the imaginary part \( b \) using \( b = r \sin \theta \).

• \( a \) and \( b \) represent projection onto the horizontal and vertical axes respectively.
• All calculations in rectangular form utilize these direct coordinates.

If given a number in polar form, calculations involve resolving the trigonometric parts before multiplying by \( r \). As shown, \( z = 10(0.809 + i \times 0.588) = 8.09 + i \times 5.88 \).

Euler's formula

Euler's formula provides a bridge between trigonometric functions and complex exponential functions. Expressed as \( e^{i\theta} = \cos \theta + i \sin \theta \), this equation allows us to encapsulate complex polar form into a more concise form.

Using Euler’s formula, the complex number \( z = r(\cos \theta + i \sin \theta) \) transforms into \( z = re^{i\theta} \). This transformation is not only elegant but simplifies complex operations like exponentiation. With \( r = 10 \) and \( \theta = \frac{\pi}{5} \), the expression becomes \( 10e^{i\frac{\pi}{5}} \).

• This reformatting is particularly advantageous in calculus and advanced algebra.
• It supports easy multiplication and division of complex numbers by simply adding or subtracting angles.

Euler's formula emphasizes the connections in mathematics, allowing seamless transitions between various mathematical concepts and formulas.

trigonometric functions

Trigonometric functions play an integral role in the conversion between different forms of complex numbers. In this context, \( \cos \theta \) and \( \sin \theta \) are used to break down the rotation and magnitude of a complex number in its polar form.

The trigonometric representation \( z = r(\cos \theta + i \sin \theta) \) uses \( \cos \theta \) for the x-coordinate and \( \sin \theta \) for the y-coordinate on the complex plane. Understanding these functions is crucial for these conversions:

• \( \cos \theta \) provides the horizontal projection of the vector.
• \( \sin \theta \) offers the vertical projection.

Calculating \( \cos \frac{\pi}{5} \approx 0.809 \) and \( \sin \frac{\pi}{5} \approx 0.588 \) in the example at hand is essential for translating a complex number from polar to rectangular form. These values help construct the real and imaginary parts when manipulating the number in calculations or visualization.