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Chapter 6: Problem 96

If you are given a complex number in rectangular form, how do you write it in polar form?

Short Answer

Expert verified
The complex number in polar form is represented as \(r (\cos 胃+ i \sin 胃)\), where \(r\) is the magnitude of the complex number and \(胃\) is the argument or angle.

Step by step solution

01

Identify real and imaginary parts

In the rectangular form, identify the real and imaginary parts of the given complex number. The real part is the number before \(i\) and the imaginary part is the number preceding \(i\).
02

Calculate magnitude (r)

The magnitude of a complex number (r) in polar form is found by taking the square root of the sum of squares of the real and imaginary parts. It is given by the formula \[r= \sqrt{{a^2+b^2}}\]
03

Calculate the argument angle (胃)

The argument (胃) of a complex number in polar form is found by taking the arctangent of the ratio of the imaginary part to the real part. It is given by the formula \[\ 胃= \arctan \frac{b}{a}\]. Note that the quadrant in which the complex number lies should be taken into account.
04

Write in Polar Form

Write the complex number in polar form using the magnitude and the argument. The polar form is written as \[r (\cos \胃+ i \sin \胃)\]

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Key Concepts

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

Rectangular Form
The rectangular form of a complex number is one of the common ways to express complex numbers. It is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
  • The real part \(a\) is the horizontal component on the complex plane.
  • The imaginary part \(b\) follows the imaginary unit \(i\) and represents the vertical component.
Visualize this form as a vector originating from the origin on a 2D plane, extending to a point \((a, b)\). Understanding this form is critical because it lays the foundation for converting to other forms, such as polar form.
Polar Form
Polar form is another way to represent complex numbers and is especially useful for multiplication and division. A complex number in polar form is expressed as \(r(\cos \theta + i\sin \theta)\), where:
  • \(r\) is the magnitude or modulus of the complex number.
  • \(\theta\) is the argument angle, known as the angle the vector makes with the positive real axis.
Polar form translates our complex number from the over-and-up "grid" view into a rotation and scale. This makes it helpful for understanding the geometric properties and operations, especially rotations and dilations.
Magnitude
The magnitude of a complex number, often referred to as \(r\), represents the length of the vector from the origin to the point \((a, b)\) in the complex plane. It can be calculated using the formula: \[r = \sqrt{a^2 + b^2}\]
  • This formula comes from the Pythagorean theorem, where the real and imaginary parts are the legs of a right triangle, and the magnitude is the hypotenuse.
  • The magnitude is always a non-negative real number.
Understanding the magnitude helps us measure the "size" of the complex number, regardless of its direction.
Argument Angle
The argument angle, denoted as \(\theta\), is the angle formed by the vector of the complex number with the positive real axis. It's calculated by:\[\theta = \arctan\left(\frac{b}{a}\right)\]However, one needs to consider the quadrant of the complex number to get the correct angle:
  • If the complex number lies in the second or third quadrant (where \(a\) is negative), we need to add \(\pi\) to \(\theta\).
  • In the fourth quadrant, you may correct the angle by adding \(2\pi\) if negative results are obtained.
This angle provides the direction of a complex number, matching the angle you might turn through if you were to rotate from the positive real axis to the vector representing your complex number.

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Most popular questions from this chapter

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=8$$

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$\left(2, \frac{\pi}{6}\right)$$

Convert each rectangular equation to a polar equation that expresses \(r\) in terms of \(\theta\) $$y^{2}=6 x$$

Polar coordinates of a point are given. Use a graphing utility to find the rectangular coordinates of each point to three decimal places. $$(5.2,1.7)$$

Solve: \(2 \sin ^{2} x-1=0,0 \leq x<2 \pi\)
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