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 21

Chapter 5, Problem 21

In Exercises 13–34, test for symmetry and then graph each polar equation. r = 1 + 2 cos θ

Verified step by step guidance

Recall the three common tests for symmetry in polar coordinates: 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 \( r = 1 + 2 \cos \theta \) and check if the equation remains unchanged.

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

To test 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 \( r \) and 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 and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and plot polar equations like r = 1 + 2 cos θ is essential for graphing curves in the polar plane.

Symmetry Tests in Polar Graphs

Testing symmetry in polar graphs involves checking if the equation remains unchanged under transformations: θ → -θ (symmetry about the polar axis), θ → π - θ (symmetry about the line θ = π/2), and r → -r with θ → θ + π (symmetry about the pole). This helps simplify graphing by identifying mirrored parts.

Graphing Polar Equations Involving Cosine

Polar equations with cosine terms, such as r = 1 + 2 cos θ, often produce limaçon shapes. Recognizing how the cosine function affects the radius for different angles allows for accurate plotting and understanding of the curve's features like loops or dimpled shapes.

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Introduction to Common Polar Equations

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 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. x = t, y = 2t

Textbook Question

In Exercises 21–28, divide and express the result in standard form. 2 / 3 - i

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 9–20, use point plotting to graph the plane curve described by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. x = 2t, y = |t − 1|; −∞ < t < ∞

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)