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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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 12.3.5
What is the slope of the line θ=π/3?
Verified step by step guidance1
Recognize that the equation \( \theta = \frac{\pi}{3} \) represents a line in polar coordinates where the angle \( \theta \) is constant at \( \frac{\pi}{3} \).
Recall that in polar coordinates, \( \theta \) is the angle measured from the positive x-axis, so the line \( \theta = \frac{\pi}{3} \) is a straight line passing through the origin making an angle of \( \frac{\pi}{3} \) with the x-axis.
To find the slope of this line in Cartesian coordinates, use the relationship between slope \( m \) and angle \( \theta \): \[ m = \tan(\theta) \].
Substitute \( \theta = \frac{\pi}{3} \) into the formula to express the slope as \( m = \tan\left(\frac{\pi}{3}\right) \).
This expression gives the slope of the line \( \theta = \frac{\pi}{3} \) in Cartesian 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 Lines
In polar coordinates, a line defined by θ = constant represents all points making a fixed angle θ with the positive x-axis. This line passes through the origin and extends outward at that angle, differing from the Cartesian form y = mx + b.
Recommended video:
05:32
Intro to Polar Coordinates
Relationship Between Angle and Slope
The slope of a line in Cartesian coordinates is the tangent of the angle it makes with the positive x-axis. Thus, for a line at angle θ, the slope m = tan(θ), linking angular direction to the line's steepness.
Recommended video:
05:45Understanding Slope Fields
Calculating Slope from Given Angle
To find the slope of the line θ = π/3, compute m = tan(π/3). Since tan(π/3) = √3, the slope of the line is √3, indicating a steep positive incline in the Cartesian plane.
Recommended video:
03:51Understanding Slope Fields Example 3
Related Practice
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
Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2.
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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
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
Textbook Question
25–30. Converting coordinates Express the following polar coordinates in Cartesian coordinates.
(4, 5π)
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