Textbook Question
40–41. {Use of Tech} Slopes of tangent lines
b. Find the slope of the lines tangent to the curve at the origin (when relevant).
r = 1 −sin θ
52
views
Ch.12 - Parametric and Polar Curves
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Not the one you use?Change textbookSelect a different textbook
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.29
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
Verified step by step guidance1
Understand the given polar equation: \(r = 3 \cos 3\theta\). This represents a polar curve where the radius \(r\) depends on the angle \(\theta\).
Recognize that the equation is of the form \(r = a \cos n\theta\), which typically produces a rose curve with petals. The number of petals depends on \(n\):
- If \(n\) is odd, the rose has \(n\) petals.
- If \(n\) is even, the rose has \$2n$ petals.
Since \(n = 3\) (an odd number), the curve will have 3 petals. The amplitude \(3\) controls the length of each petal.
To graph the curve, create a table of values by choosing several values of \(\theta\) between \(0\) and \(2\pi\), calculate the corresponding \(r\) values using \(r = 3 \cos 3\theta\), and plot the points in polar coordinates.
Connect the plotted points smoothly, noting the symmetry and petal structure, to complete the graph of the rose curve.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
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, denoted as (r, θ). Here, 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 cos 3θ.
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 or loops helps visualize curves like r = 3 cos 3θ, which produces a rose curve.
Recommended video:
3:47Introduction to Common Polar Equations
Rose Curves
Rose curves are a family of polar graphs defined by equations of the form r = a cos(nθ) or r = a sin(nθ). The number of petals depends on n: if n is odd, the curve has n petals; if even, it has 2n petals. For r = 3 cos 3θ, the graph has 3 petals.
Recommended video:
3:37Roses
Related Practice
Textbook Question
General equations for a circle Prove that the equations
X = a cos t + b sin t, y = c cos t + d sin t
where a, b, c, and d are real numbers, describe a circle of radius R provided a² +c² =b² +d² = R² and ab+cd=0.
153
views
Textbook Question
24–26. Sets in polar coordinates Sketch the following sets of points.
4 ≤ r² ≤ 9
37
views
Textbook Question
53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x = 16y²
73
views
Textbook Question
Polar conversion Write the equation r ² +r(2sinθ−6cosθ)=0 in Cartesian coordinates and identify the corresponding curve.
48
views
Textbook Question
53–57. Conic sections
c. Find the eccentricity of the curve.
y² - 4x² = 16
72
views