Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations

Problem 23

Chapter 5, Problem 23

In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3 + 4i

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Identify the complex number given: \(-3 + 4i\). Here, the real part is \(-3\) and the imaginary part is \(4\).

Plot the complex number on the complex plane, where the x-axis represents the real part and the y-axis represents the imaginary part. So, plot the point at coordinates \((-3, 4)\).

Calculate the magnitude (or modulus) \(r\) of the complex number using the formula \(r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2}\). Substitute the values to get \(r = \sqrt{(-3)^2 + 4^2}\).

Find the argument (or angle) \(\theta\) of the complex number using the formula \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right)\). Substitute the values to get \(\theta = \tan^{-1}\left(\frac{4}{-3}\right)\). Remember to consider the quadrant where the point lies to determine the correct angle.

Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the magnitude and \(\theta\) is the argument you found.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers and the Complex Plane

A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Plotting involves locating the point (a, b) on this plane.

Polar Form of Complex Numbers

Polar form represents a complex number using its magnitude (distance from origin) and argument (angle with the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r = √(a² + b²) and θ is the angle measured in degrees or radians.

Calculating the Argument (Angle)

The argument θ of a complex number is the angle formed with the positive real axis, found using θ = arctan(b/a). Care must be taken to determine the correct quadrant based on the signs of a and b, ensuring the angle accurately reflects the complex number's position.

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Related Practice

Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 2 − 3 sin θ

Textbook Question

In Exercises 22–24, find the quotient z₁/z₂ of the complex numbers. Leave answers in polar form.

z₁ = 5 (cos 4π/3 + i sin 4π/3)

z₂ = 10 (cos π/3 + i sin π/3)

Textbook Question

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

Textbook Question

In Exercises 21–28, divide and express the result in standard form.

3 / 4+i

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Textbook Question

In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i

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Textbook Question

In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)