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Chapter 8: Problem 45

In Exercises 45-60, express each complex number in exact rectangular form. $$ 5\left(\cos 180^{\circ}+i \sin 180^{\circ}\right) $$

Short Answer

Expert verified
The rectangular form is \(-5\).

Step by step solution

01

Understanding the Polar Form

The given expression is in the polar form, using the formula \( r(\cos \theta + i \sin \theta) \). Here, \( r = 5 \), and \( \theta = 180^{\circ} \). The problem is to convert this into rectangular form \( a + bi \).
02

Solving \( \cos 180^{\circ} \)

Calculate \( \cos 180^{\circ} \). From trigonometric values, \( \cos 180^{\circ} = -1 \).
03

Solving \( i \sin 180^{\circ} \)

Calculate \( i \sin 180^{\circ} \). From trigonometric values, \( \sin 180^{\circ} = 0 \). Hence, \( i \sin 180^{\circ} = 0 \cdot i = 0i \).
04

Substitute and Simplify

Substitute the values into the expression: \[5(\cos 180^{\circ} + i \sin 180^{\circ}) = 5(-1) + 5(0i) = -5 + 0i\]. Therefore, the exact rectangular form is \(-5\).

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Key Concepts

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

Polar Form
Complex numbers can be represented in different forms, one of which is the polar form. This form is denoted as \( r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude of the complex number, also called the modulus, and \( \theta \) is the angle, known as the argument. The polar form provides a geometric interpretation, where \( r \) is the distance from the origin on the complex plane, and \( \theta \) is the direction of the line joining the origin to the point. This form is especially useful in multiplication and division of complex numbers as it simplifies them using trigonometric identities.
  • The modulus \( r \) is a positive real number.
  • The argument \( \theta \) is typically measured in degrees or radians.
Rectangular Form
Rectangular form, also known as the Cartesian form, represents complex numbers as \( a + bi \). Here, \( a \) and \( b \) are real numbers, where \( a \) is the real part and \( b \) is the imaginary part. The rectangular form is straightforward and intuitive for addition and subtraction of complex numbers. It forms the basis for plotting complex numbers on the complex plane with real and imaginary axes.
  • \( a = r \cos \theta \)
  • \( b = r \sin \theta \)
These relations highlight the connection between polar and rectangular forms, enabling easy conversion between them.
Trigonometry
Trigonometric concepts are essential when discussing complex numbers, especially in their polar form. Trigonometric functions \( \cos \) and \( \sin \) help define the orientation and position of a complex number on the plane.
  • \( \cos \theta \) provides the horizontal displacement, aligning with the real part of the number.
  • \( \sin \theta \) offers the vertical displacement, corresponding to the imaginary part.
Understanding basic trigonometric values is crucial. For instance, \( \cos 180^\circ = -1 \) and \( \sin 180^\circ = 0 \), as used in the original exercise solution. These values are critical to convert between forms accurately.
Conversion Process
The conversion process involves changing a complex number from polar form to rectangular form. This transformation uses basic trigonometric functions, relying on their real number values. From the expression \( 5(\cos 180^\circ + i \sin 180^\circ) \), we follow these steps:
  • Identify \( r \) and \( \theta \) from the polar expression. Here, \( r = 5 \) and \( \theta = 180^\circ \).
  • Compute \( \cos 180^\circ = -1 \) and \( \sin 180^\circ = 0 \).
  • Substitute into the rectangular form formula, \( a = r \cos \theta \), giving \( a = 5 \times (-1) = -5 \).
  • Similarly, compute \( b = r \sin \theta = 5 \times 0 = 0 \).
Thus, the rectangular form is \( -5 + 0i \), or simply \( -5 \). This clear process ensures accurate conversion every time.

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Most popular questions from this chapter

In Exercises 81 and 82, determine whether each statement is true or false. All lima莽ons are cardioids, but not all cardioids are limacons.

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