Textbook Question
A polar conic section Consider the equation r² = sec2θ
b. Find the vertices, foci, directrices, and eccentricity of the curve."
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Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 12, Problem 12.R.31
27–32. Polar curves Graph the following equations.
r = 3 sin 4θ
Verified step by step guidance1
Recognize that the given equation is a polar equation of the form \(r = 3 \sin(4\theta)\), where \(r\) is the radius and \(\theta\) is the angle in radians.
Understand that the function \(\sin(4\theta)\) will create a rose curve with petals. The number of petals for \(r = a \sin(n\theta)\) depends on \(n\): if \(n\) is even, the curve has \$2n\( petals; if \)n\( is odd, it has \)n\( petals. Since \)n=4$ here, expect \(8\) petals.
Create a table of values by choosing several values of \(\theta\) between \(0\) and \(2\pi\), calculate \(r\) for each, and plot the points \((r, \theta)\) in polar coordinates. This helps visualize the shape of the curve.
Note the amplitude \(3\) controls the length of each petal, so the maximum radius of the petals will be \(3\). The petals will be symmetrically distributed around the origin due to the sine function and the multiple angle.
Sketch the curve by plotting the points and connecting them smoothly, ensuring the petals appear evenly spaced and reach out to radius \(3\) at their tips, forming the characteristic rose pattern.
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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 the plane using a radius and an angle (r, θ) instead of Cartesian coordinates (x, y). The radius r is the distance from the origin, and θ is the angle measured from the positive x-axis. Understanding this system is essential for graphing equations like r = 3 sin 4θ.
Recommended video:
05:32
Intro to Polar Coordinates
Graphing Polar Equations
Graphing polar equations involves plotting points by calculating r for various values of θ and then converting these to Cartesian coordinates if needed. Recognizing patterns, such as petals in rose curves, helps visualize the graph. For r = 3 sin 4θ, the number of petals and their orientation depend on the coefficient of θ.
Recommended video:
3:47Introduction to Common Polar Equations
Rose Curves
Rose curves are a family of polar graphs defined by equations like r = a sin(nθ) or r = a cos(nθ). The parameter n determines the number of petals: if n is even, the curve has 2n petals; if odd, n petals. Understanding rose curves aids in predicting the shape and symmetry of r = 3 sin 4θ.
Recommended video:
3:37Roses
Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. The polar coordinates (3, -3π/4) and (-3, π/4) describe the same point in the plane.
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Textbook Question
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1
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Textbook Question
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 8cos t + 1, y = 8sin t + 2, for 0 ≤ t ≤ 2π; t = π/3
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Textbook Question
Polar conversion Consider the equation r=4/(sinθ+cosθ).
a. Convert the equation to Cartesian coordinates and identify the curve it describes.
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Textbook Question
Jake’s response Jake responds to Liz (Exercise 33) with a graph that shows his love for her is infinite. Sketch each of the following curves. Which one should Jake send to Liz to get an infinity symbol?
b. r=(½)+sinθ
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