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Chapter 7: Problem 42

Write each complex number in the form \(a+b i\). $$8.1(\cos \pi+i \sin \pi)$$

Short Answer

Expert verified
-8.1 + 0i

Step by step solution

01

Write the complex number in polar form

The given complex number is in the polar form expressed as \[ r(\text{cos} \theta + i \text{sin} \theta) \]where \[ r = 8.1 \]and \[ \theta = \pi \].
02

Identify the real part

The real part of the complex number is given by \[ a = r \cos \theta \].Substitute the values to get:\[ a = 8.1 \cos \pi \].Using the value \cos \pi = -1, we get:\[ a = 8.1 \times -1 = -8.1 \].
03

Identify the imaginary part

The imaginary part of the complex number is given by \[ b = r \sin \theta \].Substitute the values to get:\[ b = 8.1 \sin \pi \].Using the value \sin \pi = 0, we get:\[ b = 8.1 \times 0 = 0 \].
04

Combine real and imaginary parts

Now combine the real and imaginary parts to write the complex number in the form \[ a + bi \].Thus, the complex number is:\[ -8.1 + 0i \].

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Key Concepts

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

Polar Form
The polar form of a complex number is a unique representation that expresses a complex number in terms of its magnitude (also known as the modulus) and angle (also known as the argument). It is written as \[ r (\text{cos} \theta + i \text{sin} \theta) \] where:
  • r: The magnitude or absolute value of the complex number, representing its distance from the origin in the complex plane.
  • \( \theta \): The angle that the complex number makes with the positive real axis, measured in radians.
Using the polar form can simplify many mathematical operations like multiplication and division of complex numbers. For the given exercise, the complex number was given as \[ 8.1 (\text{cos} \pi + i \text{sin} \pi) \], where r is 8.1 and \( \theta \) is \( \pi \).
Converting from polar form to the standard form a + bi requires the evaluation of the trigonometric functions cos and sin at the given angle.
Real Part
In a complex number, the real part represents the component that lies along the real axis, which can be thought of as the horizontal direction in the complex plane. When rewriting a complex number from polar form to its usual form a + bi, the real part a is calculated as \[ a = r \times \text{cos} \theta \]. In the given exercise:
  • The magnitude r is 8.1
  • The angle \( \theta \) is \( \pi \)
To find the real part, we substitute these values into the formula giving us:
\( a = 8.1 \times \text{cos} \pi \). Using the trigonometric value \( \cos \pi = -1 \), we calculate:
\( a = 8.1 \times -1 = -8.1 \). The real part of the complex number is therefore -8.1.
Imaginary Part
The imaginary part of a complex number is what lies along the imaginary axis, viewed as the vertical direction in the complex plane. It is usually given the notation bi, where b is the coefficient. For a complex number in polar form, the imaginary part b can be calculated using the formula \[ b = r \times \text{sin} \theta \]. For the given exercise:
  • The magnitude r is 8.1
  • The angle \( \theta \) is \( \pi \)
To find the imaginary part, we substitute these values into the formula resulting in:
\( b = 8.1 \times \text{sin} \pi \). Knowing that \( \sin \pi = 0 \), we calculate:
\( b = 8.1 \times 0 = 0 \). Thus, the imaginary part of the complex number is 0. Combining both real and imaginary parts results in the complex number: \( -8.1 + 0i \).

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Most popular questions from this chapter

Show that the reciprocal of \(z=r(\cos \theta+i \sin \theta)\) is \(z^{-1}=r^{-1}(\cos \theta-i \sin \theta),\) provided \(r \neq 0\)

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Show that the complex conjugate of \(z=r(\cos \theta+i \sin \theta)\) is \(\bar{z}=r(\cos (-\theta)+i \sin (-\theta))\)

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