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Ch.12 - Parametric and Polar Curves
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh.12 - Parametric and Polar CurvesProblem 12.3.31
Chapter 12, Problem 12.3.31

29–32. Intersection points Use algebraic methods to find as many intersection points of the following curves as possible. Use graphical methods to identify the remaining intersection points.


r = 1 and r = 2 sin 2θ

Verified step by step guidance
1
Start by setting the two polar equations equal to each other to find their intersection points: \(1 = 2 \sin(2\theta)\).
Solve the equation \(1 = 2 \sin(2\theta)\) for \(\sin(2\theta)\), which gives \(\sin(2\theta) = \frac{1}{2}\).
Find the general solutions for \(2\theta\) using the inverse sine function: \(2\theta = \sin^{-1}\left(\frac{1}{2}\right)\) plus all possible angles considering the periodicity of sine.
Divide the solutions for \(2\theta\) by 2 to get the corresponding values of \(\theta\) where the curves intersect.
Substitute these \(\theta\) values back into either original equation to find the corresponding \(r\) values, confirming 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 Equations

Polar coordinates represent points using a radius and an angle (r, θ) instead of Cartesian (x, y). Understanding how to interpret and manipulate polar equations like r = 1 and r = 2 sin 2θ is essential for finding intersection points between curves defined in this system.
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Intro to Polar Coordinates

Solving Equations for Intersection Points

Finding intersection points involves setting the two polar equations equal to each other and solving for θ and r. This requires algebraic manipulation and sometimes trigonometric identities to find all possible solutions where the curves meet.
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Solving Logarithmic Equations

Graphical Methods for Intersection Identification

Graphical methods involve plotting the polar curves to visually identify intersection points that may be difficult to find algebraically. This approach helps confirm solutions and locate additional intersections by analyzing the shape and symmetry of the curves.
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Euler's Method
Related Practice
Textbook Question

37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.


r = sin θ sec² θ

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Textbook Question

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


The region outside the circle r = 1/2 and inside the circle r = cos θ

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Textbook Question

Explain why the slope of the line θ=π/2 is undefined.

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Textbook Question

63–74. Arc length of polar curves Find the length of the following polar curves.


The spiral r = θ², for 0 ≤ θ ≤ 2π

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Textbook Question

39–50. Equations of ellipses and hyperbolas Find an equation of the following ellipses and hyperbolas, assuming the center is at the origin. 

A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2

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Textbook Question

23–24. Radar Airplanes are equipped with transponders that allow air traffic controllers to see their locations on radar screens. Radar gives the distance of the plane from the radar station (located at the origin) and the angular position of the plane, typically measured in degrees clockwise from north.

A plane is 50 miles from a radar station at an angle of 10 dgeree clockwise from north. Find polar coordinates for the location of the plane.

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