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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 12.2.46
Chapter 12, Problem 12.2.46

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


r = sin θ sec² θ

Verified step by step guidance
1
Recall the relationships between polar and Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, \(r^2 = x^2 + y^2\).
Start with the given equation: \(r = \sin \theta \sec^2 \theta\). Rewrite \(\sec^2 \theta\) as \(\frac{1}{\cos^2 \theta}\), so the equation becomes \(r = \sin \theta \cdot \frac{1}{\cos^2 \theta}\).
Multiply both sides of the equation by \(\cos^2 \theta\) to get \(r \cos^2 \theta = \sin \theta\).
Express \(r \cos^2 \theta\) in terms of \(x\) and \(r\): since \(x = r \cos \theta\), then \(r \cos^2 \theta = x \cos \theta\). Also, express \(\sin \theta\) as \(\frac{y}{r}\).
Substitute these into the equation to get \(x \cos \theta = \frac{y}{r}\). Then, express \(\cos \theta\) as \(\frac{x}{r}\) and substitute back to write the entire equation in terms of \(x\), \(y\), and \(r\). Finally, replace \(r\) with \(\sqrt{x^2 + y^2}\) to obtain the Cartesian form of the curve.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polar and Cartesian Coordinate Systems

Polar coordinates represent points using a radius and an angle (r, θ), while Cartesian coordinates use x and y values. Understanding how to convert between these systems is essential, using the relationships x = r cos θ and y = r sin θ.
Recommended video:
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Intro to Polar Coordinates

Trigonometric Identities

Trigonometric identities like sec θ = 1/cos θ and relationships between sine, cosine, and tangent functions help simplify expressions during conversion. Applying these identities allows rewriting polar equations in terms of x and y.
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Verifying Trig Equations as Identities

Equation Conversion and Curve Description

After substituting polar expressions with Cartesian variables, the resulting equation describes a curve in the xy-plane. Recognizing the form of this equation helps identify the type of curve, such as lines, circles, or conic sections.
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Guided course
04:47
Introduction to Parametric Equations
Related Practice
Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.


x = cos t, y = 1 + sin t; 0 ≤ t ≤ 2π

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Textbook Question

Air drop A plane traveling horizontally at 80 m/s over flat ground at an elevation of 3000 m releases an emergency packet. The trajectory of the packet is given by

x = 80t, y = −4.9t² + 3000, t ≥ 0

where the origin is the point on the ground directly beneath the plane at the moment of the release (see figure). Graph the trajectory of the packet and find the coordinates of the point where the packet lands.

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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

80–83. Equations of circles Use the results of Exercises 78–79 to describe and graph the following circles.


r² - 8r cos(θ - π/2) = 9

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Textbook Question

Cartesian lemniscate Find the equation in Cartesian coordinates of the lemniscate r² = a² cos 2θ, where a is a real number.

Textbook Question

31–36. Eliminating the parameter Eliminate the parameter to express the following parametric equations as a single equation in x and y.


x=tan t, y=sec ² t−1 

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