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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 19
Chapter 5, Problem 19

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

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Identify the complex number given, which is \(-3 + 0i\). This means the complex number lies on the real axis at \(-3\).
Plot the point on the complex plane: since the imaginary part is zero, plot the point at \(-3\) on the real axis.
Calculate the modulus (or magnitude) \(r\) of the complex number using the formula \(r = \sqrt{x^2 + y^2}\), where \(x = -3\) and \(y = 0\).
Determine the argument \(\theta\), which is the angle the line from the origin to the point makes with the positive real axis. Since the point is on the negative real axis, \(\theta\) is either \(\pi\) radians or \(180^\circ\).
Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or \(r \operatorname{cis} \theta\), substituting the values of \(r\) and \(\theta\) 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. Plotting a complex number involves representing it 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.
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Polar Form of Complex Numbers

The polar form represents a complex number using its magnitude (distance from the origin) and argument (angle with 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.
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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 to polar form.
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Related Practice
Textbook Question

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−2, − π/2)

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Textbook Question

In Exercises 15–18, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6 (cos 2π/3 + i sin 2π/3)

Textbook Question

In Exercises 19–21, find the product of the complex numbers. Leave answers in polar form.

z₁ = 3(cos 40°+i sin 40°)

z₂ = 5(cos 70°+i sin 70°)

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Textbook Question

In Exercises 11–20, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. (−1, π)

Textbook Question

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

Textbook Question

In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)²