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Chapter 8: Problem 30

In Exercises 29-44, use a calculator to express each complex number in polar form. Express Exercises 29-36 in degrees and Exercises 37-44 in radians. $$ 2+3 i $$

Short Answer

Expert verified
The polar form of the complex number is \( \sqrt{13}(\cos 56.31^\circ + i \sin 56.31^\circ) \).

Step by step solution

01

Understand the Complex Number

The given complex number is expressed in the form \( a + bi \), where \( a = 2 \) and \( b = 3 \).
02

Calculate the Magnitude

To find the magnitude (or modulus) \( r \) of the complex number \( z = 2 + 3i \), use the formula \( r = \sqrt{a^2 + b^2} \). Thus, \( r = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \).
03

Determine the Argument

The argument \( \theta \) is the angle formed with the positive real axis. It is calculated using \( \tan \theta = \frac{b}{a} \). Here, \( \tan \theta = \frac{3}{2} \). Using a calculator, \( \theta \approx 56.31^\circ \).
04

Express in Polar Form

The polar form of a complex number is given by \( z = r(\cos \theta + i \sin \theta) \). Substituting the values we get \( z = \sqrt{13}(\cos 56.31^\circ + i \sin 56.31^\circ) \).

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Key Concepts

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

Complex Numbers
Complex numbers are like a secret ingredient in mathematics that combines real and imaginary parts. These numbers are expressed as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, with \( i \) representing the imaginary unit (\( i^2 = -1 \)). In our example, \( 2 + 3i \), the number \( 2 \) is the real part and \( 3i \) is the imaginary part.
  • Real Part: The actual number part without the imaginary unit, here it is 2.
  • Imaginary Part: The part with the imaginary unit \( i \), here it is 3i.
Understanding complex numbers helps us solve equations that don鈥檛 have solutions in the real number system alone, opening up a whole new realm of possibilities.
Modulus of a Complex Number
The modulus of a complex number, sometimes called the magnitude, denotes the 'distance' from the origin to the point in the complex plane, akin to the length of a vector. In simple terms, it illustrates how 'big' the number is.To find this distance for a complex number \( a + bi \), use the formula:\[ r = \sqrt{a^2 + b^2}\]For our number \( 2 + 3i \), the calculations work out as follows:
  • Square the real part: \( a^2 = 2^2 = 4 \)
  • Square the imaginary part: \( b^2 = 3^2 = 9 \)
  • Add both squares: \( 4 + 9 = 13 \)
  • Take the square root to find the modulus: \( \sqrt{13} \approx 3.61 \)
This value tells you how far the point corresponding to the complex number is from the origin in the plane.
Argument of a Complex Number
The argument of a complex number is the angle it forms with the positive real axis in the complex plane, but in more intuitive terms, it's akin to direction. Calculating the argument gives the 'angle' direction or inclination of the complex number from the real axis.To determine the argument \( \theta \):
  • Use the formula: \( \tan \theta = \frac{b}{a} \).
  • For \( 2 + 3i \), the fraction becomes: \( \tan \theta = \frac{3}{2} \).
  • Utilize a calculator: \( \theta \approx 56.31^\circ \).
This angle can be measured in degrees or radians, helping to express where 'direction' points around the origin.
Degrees and Radians
Degrees and radians are two ways to measure angles, much like using miles and kilometers to measure distance. They are essential when working with trigonometric functions and complex numbers, particularly in converting between rectangular and polar forms.
  • Degrees: An entire circle is \( 360^\circ \).
  • Radians: An entire circle is \( 2\pi \) radians. It鈥檚 the arc length of a circle's radius.
Conversion between the two offers flexibility:
  • From degrees to radians: Multiply by \( \frac{\pi}{180} \).
  • From radians to degrees: Multiply by \( \frac{180}{\pi} \).
Understanding this can help you seamlessly switch between different modes of angular measurement, relevant for expressing complex numbers in polar form.

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Most popular questions from this chapter

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2}=16 \sin (2 \theta) $$

For Exercises 75 and 76, refer to the following: Many microphone manufacturers advertise their exceptional pickup capabilities that isolate the sound source and minimize background noise. The name of these microphones comes from the pattern formed by the range of the pickup. Cardioid Pickup Pattern. Graph the cardioid curve to see what the range of a microphone might look like: \(r=-4-4 \sin \theta\).

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=\frac{t}{2}, y=2 \tan t $$

Find the polar equation that is equivalent to a vertical line, \(x=a\).

For Exercises 77 and 78, refer to the following: The sword artistry of the samurai is legendary in Japanese folklore and myth. The elegance with which a samurai could wield a sword rivals the grace exhibited by modern figure skaters. In more modern times, such legends have been rendered digitally in many different video games (e.g., Ominusha). In order to make the characters realistically move across the screen-and in particular, wield various sword motions true to the legends-trigonometric functions are extensively used in constructing the underlying graphics module. One famous movement is a figure eight, swept out with two hands on the sword. The actual path of the tip of the blade as the movement progresses in this figure-eight motion depends essentially on the length \(L\) of the sword and the speed with which the figure is swept out. Such a path is modeled using a polar equation of the form \(r^{2}(\theta)=L \cos (A \theta)\) or \(r^{2}(\theta)=L \sin (A \theta), \theta_{1} \leq \theta \leq \theta_{2}\). Video Games. Graph the following equations: a. \(r^{2}(\theta)=5 \cos \theta, 0 \leq \theta \leq 2 \pi\) b. \(r^{2}(\theta)=5 \cos (2 \theta), 0 \leq \theta \leq \pi\) c. \(r^{2}(\theta)=5 \cos (4 \theta), 0 \leq \theta \leq \frac{\pi}{2}\) What do you notice about all of these graphs? Suppose that the movement of the tip of the sword in a game is governed by these graphs. Describe what happens if you change the domain in (b) and (c) to \(0 \leq \theta \leq 2 \pi\).
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