Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.2.109

Chapter 12, Problem 12.2.109

Subtle symmetry Without using a graphing utility, determine the symmetries (if any) of the curve r=4-sin (θ/2)

Verified step by step guidance

Recall that to determine symmetries of a polar curve \(r = f(\theta)\), we check for symmetry about the polar axis, the line \(\theta = \frac{\pi}{2}\), and the pole (origin).

For symmetry about the polar axis (the horizontal axis), replace \(\theta\) by \(-\theta\) and check if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 4 - \sin\left(\frac{-\theta}{2}\right)\) simplifies to the original \(r\).

For symmetry about the line \(\theta = \frac{\pi}{2}\) (vertical axis), replace \(\theta\) by \(\pi - \theta\) and check if the equation remains unchanged or can be manipulated to the original form. That is, check if \(r = 4 - \sin\left(\frac{\pi - \theta}{2}\right)\) simplifies to the original \(r\).

For symmetry about the pole (origin), replace \(r\) by \(-r\) and \(\theta\) by \(\theta + \pi\), and check if the equation holds. That is, check if \(-r = 4 - \sin\left(\frac{\theta + \pi}{2}\right)\) can be rearranged to the original equation.

Analyze the results from these substitutions to conclude which symmetries the curve has based on whether the equation remains equivalent after each transformation.

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Key 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 using a radius and an angle, with equations expressing relationships between r (radius) and θ (angle). Understanding how to interpret and manipulate polar equations is essential for analyzing curves defined in this system.

Symmetry in Polar Graphs

Symmetry in polar graphs can be tested by checking if the equation remains unchanged under transformations: θ replaced by -θ (symmetry about the polar axis), θ replaced by π - θ (symmetry about the vertical line θ = π/2), or r replaced by -r with θ replaced by θ + π (symmetry about the pole).

Trigonometric Function Properties

Understanding the properties of sine functions, especially with angle transformations like θ/2, helps determine how the function behaves under angle shifts or reflections. This is crucial for testing symmetry without graphing tools.

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(-4, 4√3)

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49–52. Cartesian-to-polar coordinates Convert the following equations to polar coordinates.

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93–94. Parametric equations of ellipses Find parametric equations (not unique) of the following ellipses (see Exercises 91–92). Graph the ellipse and find a description in terms of x and y.

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31–38. Equations of parabolas Find an equation of the following parabolas. Unless otherwise specified, assume the vertex is at the origin.

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37–52. Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique.

The line segment starting at P(0, 0) and ending at Q(2, 8)

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13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

x² + y²/9 = 1