Textbook Question
44–49. Areas of regions Find the area of the following regions.
The region inside the cardioid r=1+cosθ and outside the cardioid r=1−cosθ
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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.43
42–43. Intersection points Find the intersection points of the following curves.
r= √(cos3t) and r= √(sin3t)
Verified step by step guidance1
Rewrite the given curves in terms of their polar coordinates: the first curve is \(r = \sqrt{\cos(3t)}\) and the second curve is \(r = \sqrt{\sin(3t)}\).
To find the intersection points, set the two expressions for \(r\) equal to each other: \(\sqrt{\cos(3t)} = \sqrt{\sin(3t)}\).
Square both sides to eliminate the square roots, giving \(\cos(3t) = \sin(3t)\).
Use the trigonometric identity or rewrite the equation to find values of \(t\) that satisfy \(\cos(3t) = \sin(3t)\), for example by dividing both sides by \(\cos(3t)\) (where defined) to get \(1 = \tan(3t)\), so \(\tan(3t) = 1\).
Solve for \(t\) by finding all angles where \(\tan(3t) = 1\), then substitute these \(t\) values back into either \(r = \sqrt{\cos(3t)}\) or \(r = \sqrt{\sin(3t)}\) to find the corresponding \(r\) values, which together give the intersection points in polar coordinates.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polar Coordinates and Parametric Curves
Polar coordinates represent points using a radius and angle (r, t), where r is the distance from the origin and t is the angle. Understanding how curves are defined in polar form, especially with trigonometric functions of multiples of t, is essential to analyze and find intersections.
Recommended video:
05:32
Intro to Polar Coordinates
Solving Equations Involving Trigonometric Functions
Finding intersection points requires solving equations where expressions involving trigonometric functions like cos(3t) and sin(3t) are equal. Knowledge of trigonometric identities and properties helps simplify and solve these equations for t.
Recommended video:
6:04Introduction to Trigonometric Functions
Conditions for Intersection in Polar Coordinates
Two polar curves intersect where their radius values are equal for the same angle or where their points coincide in the plane. This involves equating r-values and considering the domain restrictions (e.g., square roots require non-negative arguments) to find valid intersection points.
Recommended video:
05:32Intro to Polar Coordinates
Related Practice
Textbook Question
7–8. Parametric curves and tangent lines
a. Eliminate the parameter to obtain an equation in x and y.
x = 4sin 2t, y = 3cos 2t, for 0 ≤ t ≤ π; t = π/6
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Textbook Question
10–12. Parametric curves
a. Eliminate the parameter to obtain an equation in x and y.
x = ln t, y = 8ln t², for 1 ≤ t ≤ e²; (1, 16)
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Textbook Question
24–26. Sets in polar coordinates Sketch the following sets of points.
4 ≤ r² ≤ 9
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Textbook Question
61–64. Polar equations for conic sections Graph the following conic sections, labeling vertices, foci, directrices, and asymptotes (if they exist). Give the eccentricity of the curve. Use a graphing utility to check your work.
r = 3/(1 - 2 cos θ)
Textbook Question
22–23. Arc length Find the length of the following curves.
x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4