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 33

Chapter 5, Problem 33

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

Verified step by step guidance

Rewrite the given polar equation \(r \cos \theta = -3\) in a more recognizable form. Recall that \(r \cos \theta\) represents the \(x\)-coordinate in Cartesian coordinates, so the equation can be rewritten as \(x = -3\).

Analyze the symmetry of the equation with respect to the polar axes: the polar axis (the positive \(x\)-axis), the line \(\theta = \frac{\pi}{2}\) (the vertical axis), and the pole (origin).

Test for symmetry about the polar axis by replacing \(\theta\) with \(-\theta\) in the original equation and checking if the equation remains unchanged.

Test for symmetry about the line \(\theta = \frac{\pi}{2}\) by replacing \(\theta\) with \(\pi - \theta\) and checking if the equation remains unchanged.

Test for symmetry about the pole by replacing \(r\) with \(-r\) and \(\theta\) with \(\theta + \pi\) and checking if the equation remains unchanged. Then, sketch the graph of the line \(x = -3\) in the Cartesian plane, which corresponds to a vertical line 3 units to the left of the origin.

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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 relationships between r and θ, describing curves in the plane. Understanding how to interpret and manipulate these equations is essential for graphing and analyzing their properties.

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. Identifying symmetry helps simplify graphing and understanding the curve's shape.

Conversion Between Polar and Cartesian Coordinates

Converting polar equations to Cartesian form using x = r cos θ and y = r sin θ can simplify analysis. For example, r cos θ = x, so the given equation can be rewritten in Cartesian coordinates to identify the curve type. This conversion aids in graphing and understanding the geometric nature of the equation.

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Intro to Polar Coordinates

Related Practice

Textbook Question

In Exercises 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point. (4, 90°)

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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 = 2 + 4 cos t, y = −1 + 3 sin t; 0 ≤ t ≤ π

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex cube roots of 8i

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Textbook Question

In Exercises 27–32, select the representations that do not change the location of the given point. (−6, 3π) (6, −π)

Textbook Question

In Exercises 32–35, find all the complex roots. Write roots in rectangular form. The complex fourth roots of 16 (cos 2π/3 + i sin 2π/3)

Textbook Question

In Exercises 27–32, select the representations that do not change the location of the given point. (−5, − π/4) (−5, 7π/4)