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 15

Chapter 5, Problem 15

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

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Recall the three common tests for symmetry in 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 for symmetry about the polar axis, replace \( \theta \) with \( -\theta \) in the equation \( r = 1 - \sin \theta \) and check if the equation remains unchanged.

To test for symmetry about the line \( \theta = \frac{\pi}{2} \), replace \( \theta \) with \( \pi - \theta \) and check if the equation remains unchanged.

To test for symmetry about the pole, replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) and check if the equation remains unchanged.

After determining the symmetries, plot points for various values of \( \theta \) (for example, \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{2}, \pi \), etc.) by calculating corresponding \( r \) values using \( r = 1 - \sin \theta \), then sketch the graph accordingly.

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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.

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 simplify graphing and understand the curve's shape.

Graphing Polar Equations Involving Sine

Polar equations with sine functions, like r = 1 − sin θ, often produce limaçon or cardioid shapes. Recognizing the effect of the sine term on the radius as θ varies helps in sketching the curve accurately, noting key points such as maximum and minimum r values.

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Related Practice

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. −4i

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Textbook Question

In Exercises 9–20, find each product and write the result in standard form.

(3 + 5i)(3 − 5i)

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

In Exercises 9–20, find each product and write the result in standard form. (−5 + i)(−5 − i)

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. (3, 4π/3)

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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