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 =3 − 6 cos θ
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.11a
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = 3cos(-t), y = 3sin(-t) - 1, for 0 ≤ t ≤ π; (0, -4)
Verified step by step guidance1
Recognize that the given parametric equations are \(x = 3\cos(-t)\) and \(y = 3\sin(-t) - 1\) with the parameter \(t\) in the interval \(0 \leq t \leq \pi\).
Recall the trigonometric identities for cosine and sine of negative angles: \(\cos(-t) = \cos t\) and \(\sin(-t) = -\sin t\). Use these to rewrite the parametric equations as \(x = 3\cos t\) and \(y = -3\sin t - 1\).
Isolate the trigonometric functions from the parametric equations: \(\cos t = \frac{x}{3}\) and \(\sin t = -\frac{y + 1}{3}\).
Use the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\) to eliminate the parameter \(t\). Substitute the expressions for \(\sin t\) and \(\cos t\) into this identity:
\[\left(\frac{x}{3}\right)^2 + \left(-\frac{y + 1}{3}\right)^2 = 1.\]
Simplify the equation to obtain a relation purely in terms of \(x\) and \(y\), which represents the Cartesian equation of the curve without the parameter \(t\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parametric Equations
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
Recommended video:
Guided course
08:02
Parameterizing Equations
Eliminating the Parameter
Eliminating the parameter involves manipulating the parametric equations to remove the parameter t, resulting in a direct relationship between x and y. This often requires using trigonometric identities or algebraic techniques to rewrite the curve in Cartesian form.
Recommended video:
Guided course
05:59Eliminating the Parameter
Trigonometric Identities
Trigonometric identities, such as sin(-t) = -sin(t) and cos(-t) = cos(t), are essential for simplifying parametric equations involving sine and cosine. These identities help transform and combine expressions to eliminate the parameter and find the Cartesian equation.
Recommended video:
7:17Verifying Trig Equations as Identities
Related Practice
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 (±4, 0) and directrices x = ±2
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
Textbook Question
27–32. Polar curves Graph the following equations.
r = 3 cos 3θ
1
views
Textbook Question
19–20. Area bounded by parametric curves Find the area of the following regions. (Hint: See Exercises 103–105 in Section 12.1.) The region bounded by the y-axis and the parametric curve
The region bounded by the x-axis and the parametric curve x=cost, y=sin2t, for 0≤t≤π/2
1
views
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
1
views