Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.R.31

Chapter 12, Problem 12.R.31

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ

Verified step by step guidance

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.

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.

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.

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Roses

Related Practice

Textbook Question

25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.

(2, 7π/4)

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Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region inside the lemniscate r² = 6 sin 2θ

Textbook Question

65–68. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Graph the curve, labeling vertices, foci, asymptotes (if they exist), and directrices.

A hyperbola with vertices (0, ±2) and directrices y = ±1

Textbook Question

53–57. Conic sections

c. Find the eccentricity of the curve.

x²/4 + y²/25 = 1

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Textbook Question

3–6. Eliminating the parameter Eliminate the parameter to find a description of the following curves in terms of x and y. Give a geometric description and the positive orientation of the curve.

x = sin t - 3, y = cos t + 6; 0 ≤ t ≤ π

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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

27–32. Polar curves Graph the following equations.r = 3 sin 4θ

Verified video answer for a similar problem:

Key Concepts

Polar Coordinates

Graphing Polar Equations

Rose Curves

27–32. Polar curves Graph the following equations.

r = 3 sin 4θ