Problem 31 Carry out the indicated operatio... [FREE SOLUTION]

Chapter 13: Problem 31

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\sqrt{2}\left(\cos \frac{1}{3} \pi+i \sin \frac{1}{3} \pi\right) \times \sqrt{2}\left(\cos \frac{4}{3} \pi+i \sin \frac{4}{3} \pi\right)$$

Short Answer

Expert verified

The result is -2.

Step by step solution

Convert to Rectangular Form

To solve the expression, first realize that each term is written in the form of a complex number in polar form: $r(\cos \theta + i \sin \theta)$. We can convert these to rectangular form. Recall Euler's Formula: $e^{i\theta} = \cos \theta + i \sin \theta$. By using this, we can write $\sqrt{2}(\cos \frac{1}{3} \pi + i \sin \frac{1}{3} \pi)$ and $\sqrt{2}(\cos \frac{4}{3} \pi + i \sin \frac{4}{3} \pi)$ as equivalent to $\sqrt{2}e^{i\frac{1}{3}\pi}$ and $\sqrt{2}e^{i\frac{4}{3}\pi}$.

Multiply Complex Numbers in Polar Form

To multiply two complex numbers in polar form $r_1e^{i\theta_1}$ and $r_2e^{i\theta_2}$, multiply their magnitudes and add their angles: $r_1r_2e^{i(\theta_1 + \theta_2)}$. Applying this, we have \[\sqrt{2} \times \sqrt{2} e^{i(\frac{1}{3} \pi + \frac{4}{3} \pi)} = 2e^{i\pi} . \]

Convert Back to Rectangular Form

The result from the multiplication is $2e^{i\pi}$. Using Euler鈥檚 formula this converts to rectangular form as $2(\cos \pi + i \sin \pi) $. Evaluating the trigonometric functions, $\cos \pi = -1$ and $\sin \pi = 0$, thus: \[2(-1 + i0) = -2. \]

Final Result

The complex product of the initial terms, expressed in rectangular form, is $-2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Form of Complex Numbers

Complex numbers can be represented in multiple ways, and one of the very expressive methods is the polar form. In polar representation, a complex number is shown as the combination of a magnitude (or modulus) and an angle (or argument). This is helpful, especially for mathematical operations like multiplication or division.

A complex number in polar form is written as $r(\cos \theta + i \sin \theta)$, where $r$ is the magnitude and $\theta$ is the angle.
The magnitude $r$ is calculated as $r = \sqrt{x^2 + y^2}$, where $x$ and $y$ are the real and imaginary components, respectively.
The angle $\theta$ is given by $\tan^{-1}(y/x)$.

Polar form is incredibly useful when performing operations such as multiplication and division because it simplifies the arithmetic to multiplication of magnitudes and addition of angles.

Rectangular Form of Complex Numbers

While the polar form gives us a geometrical perspective, the rectangular form or Cartesian form displays a complex number as a straightforward combination of its real and imaginary parts: $x + yi$. This form is often more intuitive because it resembles a vector in a coordinate system.

The real part $x$ is the horizontal component, parallel to the x-axis.
The imaginary part $yi$ is the vertical component, parallel to the y-axis.

Converting from polar to rectangular form requires the use of trigonometric functions to find the real and imaginary components:\[x = r \cos \theta\]\[y = r \sin \theta\]This conversion helps to simplify and solve mathematical expressions, as one can directly handle and manipulate separate real and imaginary parts with ease.

Euler's Formula

Euler's formula is a fascinating bridge between exponential functions and trigonometric functions. It transforms the polar form into an exponential form and is written as $e^{i\theta} = \cos \theta + i \sin \theta$. This elegant expression simplifies handling complex numbers, especially when performing multiplication and division in polar form.

The formula itself emerges from the complex exponential function, where $i$ is the imaginary unit ($i^2 = -1$).
By expressing polar coordinates using Euler's formula, many mathematical operations can be simplified significantly.

In context to the problem, Euler's formula was employed to convert polar representations into a more manageable form for multiplication: $\sqrt{2}e^{i\frac{1}{3}\pi}$ and $\sqrt{2}e^{i\frac{4}{3}\pi}$. Then, multiplying these gives $2e^{i\pi}$, which is readily convertible back to rectangular form.

Trigonometric Functions in Complex Numbers

Trigonometric functions are essential when working with complex numbers, particularly in converting between polar and rectangular forms. They provide the means to translate an angle into horizontal and vertical components, which are the real and imaginary parts of a complex number.

$\cos \theta$ gives the x-component and is critical in finding the real part of the complex number.
$\sin \theta$ gives the y-component and helps determine the imaginary part.
Common angles ($\pi/3$, $\pi/4$, $\pi$, etc.) often do not require calculators as their trigonometric values are well-known.

In our original problem, evaluating $\cos \pi$ and $\sin \pi$ converts the final expression from polar to rectangular form. $\cos \pi = -1$ and $\sin \pi = 0$, leading to the solution of $-2$. Understanding these simple values is key to efficiently converting between different forms of complex numbers.

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Short Answer

Step by step solution

Convert to Rectangular Form

Multiply Complex Numbers in Polar Form

Convert Back to Rectangular Form

Final Result

Key Concepts

Polar Form of Complex Numbers

Rectangular Form of Complex Numbers

Euler's Formula

Trigonometric Functions in Complex Numbers

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