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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 43
Chapter 5, Problem 43

In Exercises 35–44, test for symmetry and then graph each polar equation. r = 2 + 3 sin 2θ

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Recall the three types of symmetry tests for polar equations: symmetry about the polar axis (x-axis), symmetry about the line \( \theta = \frac{\pi}{2} \) (y-axis), and symmetry about the pole (origin).
To test symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation and check if the equation remains unchanged. For \( r = 2 + 3 \sin 2\theta \), substitute \( -\theta \) to get \( r = 2 + 3 \sin(-2\theta) \).
To test symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged. Substitute \( \pi - \theta \) into the equation to get \( r = 2 + 3 \sin 2(\pi - \theta) \).
To test symmetry about the pole (origin), replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \), then check if the equation remains unchanged. Substitute these into the equation to get \( -r = 2 + 3 \sin 2(\theta + \pi) \).
After determining the symmetries, sketch the graph by plotting points for various values of \( \theta \) between 0 and \( 2\pi \), calculating corresponding \( r \) values using the equation \( r = 2 + 3 \sin 2\theta \), and then plotting these points in polar coordinates.

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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

Trigonometric Functions in Polar Equations

Polar equations often include trigonometric functions like sine and cosine, which influence the shape and periodicity of the graph. Recognizing how functions like sin(2θ) affect the curve's lobes and symmetry is crucial for accurate graphing and analysis.
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Introduction to Common Polar Equations
Related Practice
Textbook Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. (−2, 2)

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In Exercises 37–52, perform the indicated operations and write the result in standard form. ___ −8 + √−32 / 24

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Textbook Question

In Exercises 41–48, the rectangular coordinates of a point are given. Find polar coordinates of each point. Express θ in radians. _ (2,−2√3)

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In Exercises 65–68, find all the complex roots. Write roots in polar form with θ in degrees. The complex cube roots of 8(cos 210° + i sin 210°)

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In Exercises 64–70, graph each polar equation. Be sure to test for symmetry.r = 2 + cos θ
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Textbook Question

In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−3 − √−7)²