Textbook Question
Without calculating derivatives, determine the slopes of each of the lines tangent to the curve r=8 cos θ−4 at the origin.
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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 12.2.43
37–48. Polar-to-Cartesian coordinates Convert the following equations to Cartesian coordinates. Describe the resulting curve.
r = 6 cos θ + 8 sin θ
Verified step by step guidance1
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 polar equation: \(r = 6 \cos \theta + 8 \sin \theta\).
Multiply both sides of the equation by \(r\) to eliminate the denominator when substituting: \(r \cdot r = r (6 \cos \theta + 8 \sin \theta)\), which gives \(r^2 = 6r \cos \theta + 8r \sin \theta\).
Substitute \(r^2\) with \(x^2 + y^2\), \(r \cos \theta\) with \(x\), and \(r \sin \theta\) with \(y\) to get: \(x^2 + y^2 = 6x + 8y\).
Rewrite the equation to standard form by bringing all terms to one side: \(x^2 - 6x + y^2 - 8y = 0\). Then, complete the square for both \(x\) and \(y\) terms to identify the type and properties 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, y) positions on a plane. Understanding how to convert between these systems is essential for analyzing curves defined in polar form.
Recommended video:
05:32
Intro to Polar Coordinates
Conversion Formulas Between Polar and Cartesian Coordinates
The key formulas for conversion are x = r cos θ and y = r sin θ. Additionally, r² = x² + y² and tan θ = y/x. These relationships allow rewriting polar equations in terms of x and y to identify the curve in Cartesian form.
Recommended video:
05:32Intro to Polar Coordinates
Identifying and Describing Curves from Equations
After converting to Cartesian form, recognizing the type of curve (e.g., circle, ellipse, line) involves rearranging the equation and comparing it to standard forms. This helps in describing the geometric shape represented by the original polar equation.
Recommended video:
6:50Convert Equations from Polar to Rectangular
Related Practice
Textbook Question
Tangent line at the origin Find the polar equation of the line tangent to the polar curve r=4cosθ at the origin. Explain why the slope of this line is undefined.
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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.
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Textbook Question
What is the polar equation of the horizontal line y = 5?
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Textbook Question
15–22. Sets in polar coordinates Sketch the following sets of points.
2 ≤ r ≤ 8
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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=2 sin 8t, y=2 cos 8t
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