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 67

Chapter 5, Problem 67

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

Verified step by step guidance

Identify the type of polar equation: The given equation \( r = 2 + \cos \theta \) is a limaçon, which is a type of polar curve.

Test for symmetry: Check for symmetry with respect to the polar axis (x-axis), the line \( \theta = \frac{\pi}{2} \) (y-axis), and the pole (origin). For this equation, substitute \( \theta \) with \( -\theta \) to test for x-axis symmetry, and substitute \( r \) with \(-r\) to test for origin symmetry.

Determine key points: Calculate \( r \) for key angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) to find specific points on the graph.

Plot the points: Use the calculated points to plot the graph on polar coordinates. Remember that the graph will be symmetric based on the tests conducted earlier.

Sketch the curve: Connect the plotted points smoothly to form the limaçon shape, ensuring to reflect any symmetries identified in the earlier steps.

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Key Concepts

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

Polar Coordinates

Polar coordinates represent points in a plane using a distance from a reference point (the pole) and an angle from a reference direction. In the equation r = 2 + cos θ, 'r' denotes the radius or distance from the origin, while 'θ' is the angle measured from the positive x-axis. Understanding how to convert between polar and Cartesian coordinates is essential for graphing polar equations.

Symmetry in Polar Graphs

Symmetry in polar graphs can be determined by analyzing the equation with respect to specific angles. A polar graph is symmetric about the polar axis if replacing θ with -θ yields the same equation, and it is symmetric about the line θ = π/2 if replacing θ with π - θ does so. Testing for symmetry helps in sketching the graph accurately and understanding its properties.

Graphing Polar Equations

Graphing polar equations involves plotting points based on the values of 'r' for various angles 'θ'. The equation r = 2 + cos θ describes a limaçon shape, which can exhibit different characteristics based on the coefficients involved. To graph it effectively, one must calculate 'r' for key angles (like 0, π/2, π, and 3π/2) and then plot these points in the polar coordinate system.

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