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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 23
Chapter 5, Problem 23

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

Verified step by step guidance
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Recall that to test for symmetry in polar equations, we check three types of symmetry: symmetry about the polar axis (the horizontal axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (the vertical axis), and symmetry about the pole (the origin).
To test symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and see if the equation remains unchanged. For the given equation \( r = 2 - 3 \sin \theta \), substitute \( \theta \) with \( -\theta \) to get \( r = 2 - 3 \sin(-\theta) \).
Recall that \( \sin(-\theta) = -\sin \theta \), so the equation becomes \( r = 2 + 3 \sin \theta \). Since this is not the same as the original equation, the graph is not symmetric about the polar axis.
To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \theta \) with \( \pi - \theta \) in the original equation: \( r = 2 - 3 \sin(\pi - \theta) \).
Use the identity \( \sin(\pi - \theta) = \sin \theta \), so the equation becomes \( r = 2 - 3 \sin \theta \), which is the same as the original. Therefore, the graph is symmetric about the line \( \theta = \frac{\pi}{2} \).

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Key Concepts

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

Polar Coordinates and Equations

Polar coordinates represent points using a radius r and an angle θ from the positive x-axis. Polar equations express r as a function of θ, describing curves in the plane. Understanding how to interpret and plot these equations is essential for graphing polar curves.
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Intro to Polar Coordinates

Symmetry Tests in Polar Graphs

Symmetry in polar graphs can be tested about the polar axis, the line θ = π/2, and the pole (origin). These tests involve substituting θ with -θ, π - θ, or replacing r with -r to check if the equation remains unchanged, helping to identify symmetrical properties of the curve.
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Cardioids

Graphing Polar Equations Involving Sine

Polar equations with sine functions, like r = 2 − 3 sin θ, often produce limaçon or cardioid shapes. Recognizing the role of sine in shifting and shaping the curve aids in sketching the graph accurately by evaluating r at key angles and understanding the curve's behavior.
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Introduction to Common Polar Equations
Related Practice
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 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

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

2
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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.


(5, π/6)

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)