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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook
All textbooksBlitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblem 15
Chapter 5, Problem 15

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. −4i

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1
Identify the complex number given: \(-4i\). This means the real part is 0 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. Since the real part is 0 and the imaginary part is \(-4\), plot the point at \((0, -4)\).
Calculate the magnitude (or modulus) \(r\) of the complex number using the formula \(r = \sqrt{a^2 + b^2}\), where \(a\) is the real part and \(b\) is the imaginary part. Here, \(r = \sqrt{0^2 + (-4)^2}\).
Determine the argument (or angle) \(\theta\) of the complex number. Since the point lies on the negative imaginary axis, the argument is the angle measured from the positive real axis to the point \((0, -4)\). Express \(\theta\) in degrees or radians accordingly.
Write the complex number in polar form using the formula \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), where \(r\) is the magnitude and \(\theta\) is the argument found in the previous steps.

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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. The complex plane represents these numbers graphically, with the horizontal axis for the real part and the vertical axis for the imaginary part. Plotting a complex number involves locating the point (a, b) on this plane.
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Polar Form of Complex Numbers

Polar form expresses a complex number using its magnitude (distance from the origin) and argument (angle from the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r is the modulus and θ is the argument in degrees or radians. This form highlights the geometric interpretation of complex numbers.
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Complex Numbers In Polar Form

Calculating Magnitude and Argument

The magnitude r of a complex number a + bi is found using r = √(a² + b²). The argument θ is the angle formed with the positive real axis, calculated using θ = arctan(b/a), adjusted for the correct quadrant. These values are essential for converting from rectangular to polar form.
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Related Practice
Textbook Question

Test for symmetry and then graph each polar equation. r = 2 cos θ

Textbook Question

Eliminate the parameter and graph the plane curve represented by the parametric equations. Use arrows to show the orientation of each plane curve. x = √t , y = t + 1; −∞ < t < ∞

Textbook Question

Use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (3, 90°)

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Textbook Question

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 − sin θ

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Textbook Question

Graph each polar equation. r = 1 + sin θ

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Textbook Question

Write each complex number in rectangular form. If necessary, round to the nearest tenth. 8(cos 60° + i sin 60°)

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