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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 12.2.77d
Chapter 12, Problem 12.2.77d

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.  


d. The point (3,π/2) lies on the graph of r=3 cos 2θ.  

Verified step by step guidance
1
Recall that the graph is given in polar coordinates by the equation \(r = 3 \cos 2\theta\).
To check if the point \((3, \frac{\pi}{2})\) lies on the graph, substitute \(\theta = \frac{\pi}{2}\) into the equation to find the corresponding \(r\) value.
Calculate \(r\) by evaluating \(3 \cos \left( 2 \times \frac{\pi}{2} \right) = 3 \cos \pi\).
Recall that \(\cos \pi = -1\), so \(r = 3 \times (-1) = -3\).
Since the given point has \(r = 3\) but the equation yields \(r = -3\) at \(\theta = \frac{\pi}{2}\), the point \((3, \frac{\pi}{2})\) does not lie on the graph.

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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 points in the plane using a radius and an angle (r, θ), where r is the distance from the origin and θ is the angle from the positive x-axis. Understanding how to interpret and plot points in polar form is essential for analyzing whether a point lies on a given polar curve.
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Intro to Polar Coordinates

Polar Equations and Graphs

A polar equation expresses the radius r as a function of the angle θ, such as r = 3 cos 2θ. To determine if a point lies on the graph, substitute the given θ into the equation and check if the resulting r matches the point's radius. This process verifies the point's membership on the curve.
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Introduction to Common Polar Equations

Trigonometric Function Evaluation

Evaluating trigonometric functions like cosine at specific angles is crucial for solving polar equations. For example, calculating cos(2θ) when θ = π/2 requires knowledge of angle doubling and cosine values, which helps determine the correct radius and verify if the point satisfies the equation.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

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

d. Describe the curve.


x=−t+6, y=3t−3; −5≤t≤5 

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

d. The point on a parabola closest to the focus is the vertex.

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Textbook Question

11–14. Working with parametric equations Consider the following parametric equations.

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

d. Describe the curve.


x=2 t,y=3t−4;−10≤t≤10 

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Textbook Question

Intersecting lines Consider the following pairs of lines. Determine whether the lines are parallel or intersecting. If the lines intersect, then determine the point of intersection.


c. x = 1 + 3s, y = 4 + 2s and x = 4 - 3t, y = 6 + 4t

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


d. The parametric equations x=cos t, y=sin t, for −π/2≤t≤π/2, describe a semicircle.

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Textbook Question

(Use of Tech) Finger curves: r = f(θ) = cos(aᶿ) - 1.5, where a = (1 + 12π)^(1/(2π)) ≈ 1.78933

d. Plot the curve with various values of k. How many fingers can you produce?