Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
1 < r < 2 and π/6 ≤ θ ≤ π/3
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16. Parametric Equations & Polar Coordinates
Polar Coordinates
3:16 minutesProblem 12.2.58
Textbook Question
57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.
r = 2 - 2 sin θ b
Verified step by step guidance1
Identify the given polar equation: \(r = 2 - 2 \sin \theta\). This represents a polar curve where \(r\) depends on the angle \(\theta\).
Recall that \(r\) is the distance from the origin to a point on the curve, and \(\theta\) is the angle measured from the positive x-axis. To graph, you will plot points for various values of \(\theta\) between \$0$ and \(2\pi\).
Create a table of values by choosing several \(\theta\) values (for example, \$0\(, \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{2}\), \(\pi\), etc.) and compute the corresponding \)r$ values using the equation \(r = 2 - 2 \sin \theta\).
Plot each point in polar coordinates by moving \(r\) units from the origin at the angle \(\theta\). Connect these points smoothly to reveal the shape of the curve.
Use a graphing utility to input the equation \(r = 2 - 2 \sin \theta\) and compare the plotted points and shape with your manual graph to verify accuracy.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Polar Equations
Polar coordinates represent points in the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). Understanding how to interpret and plot equations given in polar form, like r = 2 - 2 sin θ, is essential for graphing polar curves.
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Intro to Polar Coordinates
Graphing Polar Curves
Graphing polar curves involves plotting points for various values of θ and connecting them smoothly. Recognizing common shapes such as cardioids, limacons, and circles helps in sketching the curve accurately before verifying with a graphing utility.
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Use of Graphing Utilities
Graphing utilities, such as graphing calculators or software, allow for precise visualization of polar curves. They help confirm the shape and key features of the graph, ensuring accuracy and aiding in understanding complex polar equations.
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