Textbook Question
How does the eccentricity determine the type of conic section?
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 12.2.30
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(4, 5π)
Verified step by step guidance1
Recall the formulas to convert polar coordinates \((r, \theta)\) to Cartesian coordinates \((x, y)\):
\(x = r \cos(\theta)\)
\(y = r \sin(\theta)\)
Identify the given polar coordinates: \(r = 4\) and \(\theta = 5\pi\).
Substitute the values into the conversion formulas:
\(x = 4 \cos(5\pi)\)
\(y = 4 \sin(5\pi)\)
Use the periodic properties of trigonometric functions to simplify \(\cos(5\pi)\) and \(\sin(5\pi)\). Remember that \(\cos(\theta)\) and \(\sin(\theta)\) have periods of \(2\pi\).
Calculate the simplified values of \(x\) and \(y\) to express the Cartesian coordinates corresponding to the given 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
Polar coordinates represent a point in the plane using a radius and an angle, denoted as (r, θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. Understanding this system is essential for converting to Cartesian coordinates.
Recommended video:
05:32
Intro to Polar Coordinates
Conversion Formulas from Polar to Cartesian
To convert polar coordinates (r, θ) to Cartesian coordinates (x, y), use the formulas x = r cos(θ) and y = r sin(θ). These formulas translate the radius and angle into horizontal and vertical distances on the Cartesian plane.
Recommended video:
3:37Convert Equations from Rectangular to Polar
Angle Measurement and Trigonometric Values
Angles in polar coordinates are often given in radians. Knowing how to evaluate trigonometric functions like sine and cosine at specific angles, such as multiples of π, is crucial for accurate conversion to Cartesian coordinates.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
69–72. Tangent lines Find an equation of the line tangent to the following curves at the given point.
y² - x²/64 = 1; (6, -5/4)
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Textbook Question
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 = 2 cos θ and r = 1 + cos θ
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Textbook Question
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
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Textbook Question
What is the slope of the line θ=π/3?
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Textbook Question
53–56. Eccentricity-directrix approach Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work.
An ellipse with vertices (0, ±9) and eccentricity ¼