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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 12.1.89d
Chapter 12, Problem 12.1.89d

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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1
Recall that the parametric equations \(x = \cos t\) and \(y = \sin t\) describe a point on the unit circle for any real number \(t\), because \(\cos^2 t + \sin^2 t = 1\) holds for all \(t\).
The parameter \(t\) typically represents the angle measured from the positive x-axis, moving counterclockwise around the circle.
Given the interval \(-\frac{\pi}{2} \leq t \leq \frac{\pi}{2}\), identify which portion of the unit circle these values of \(t\) cover. This interval corresponds to angles starting from \(-90^\circ\) to \(90^\circ\).
Visualize or sketch the unit circle and mark the points corresponding to \(t = -\frac{\pi}{2}\) and \(t = \frac{\pi}{2}\). Notice that these points lie on the lower and upper parts of the circle, respectively, and the path traced moves along the right half of the circle.
Conclude whether the curve described by these parametric equations over the given interval is a semicircle, and specify which semicircle it represents (left or right half).

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Key Concepts

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

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. Instead of y as a function of x, both x and y depend on t, allowing the description of more complex curves like circles or ellipses.
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Parameterizing Equations

Unit Circle and Trigonometric Functions

The equations x = cos t and y = sin t trace points on the unit circle as t varies. Since cosine and sine represent the x and y coordinates of a point on the circle, these parametric equations describe a circular path with radius 1.
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Evaluate Composite Functions - Values on Unit Circle

Domain of the Parameter and Curve Segment

The interval for the parameter t determines which part of the curve is traced. For t between −π/2 and π/2, the parametric equations trace only half of the unit circle, specifically the right semicircle, because cosine is non-negative in this range.
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Related Practice
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

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.  

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Textbook Question

Tangents and normals: Let a polar curve be described by r = f(θ), and let ℓ be the line tangent to the curve at the point P(x,y) = P(r,θ) (see figure).

e. Prove that the values of θ for which ℓ is parallel to the y-axis satisfy tan θ = f(θ)/f'(θ).


Textbook Question

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


e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

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

(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?