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Ch. 11 - Parametric Equations and Polar Coordinates
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 11 - Parametric Equations and Polar CoordinatesProblem 11.1.24
Chapter 11, Problem 11.1.24

Finding Cartesian from Parametric Equations


In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.


x = cos t, y = sin 3t


Graphs of three parametric curves labeled D, E, and F, showing a circle, a spiral, and a wave-like pattern.

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Step 1: Identify the parametric equations given: \(x = \cos t\) and \(y = \sin 3t\).
Step 2: Understand the behavior of \(x = \cos t\): it oscillates between -1 and 1 with period \(2\pi\).
Step 3: Understand the behavior of \(y = \sin 3t\): it oscillates between -1 and 1 but with a period of \(\frac{2\pi}{3}\), which is three times faster than \(x\).
Step 4: Look for a graph where the \(x\)-values oscillate smoothly between -1 and 1, while the \(y\)-values oscillate more rapidly, creating multiple waves within one period of \(x\). This will create a pattern with three oscillations in \(y\) for every one oscillation in \(x\).
Step 5: Match this behavior to the graphs provided. The graph labeled F shows a wave pattern with multiple oscillations in \(y\) for each oscillation in \(x\), consistent with \(x = \cos t\) and \(y = \sin 3t\).

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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, usually denoted as t. Instead of y as a function of x, both x and y depend on t, allowing representation of more complex curves like loops or spirals.
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Conversion to Cartesian Form

Converting parametric equations to Cartesian form involves eliminating the parameter t to find a direct relationship between x and y. This helps identify the shape of the curve and compare it to standard graphs, such as circles or ellipses.
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Trigonometric Parametric Curves

When parametric equations involve trigonometric functions like sine and cosine, the resulting curves often represent periodic or oscillatory shapes. Understanding how frequencies and amplitudes affect the curve is essential for matching equations to their graphs.
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Related Practice
Textbook Question

Centroids


Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

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Textbook Question

Finding Lengths of Polar Curves


Find the lengths of the curves in Exercises 21–28.


The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

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Textbook Question

Identifying Graphs


Match the parabolas in Exercises 1−4 with the following equations: x² = 2y, x² = −6y, y² = 8x, y² = −4x


Then find each parabola's focus and directrix.



Textbook Question

Ellipses and Eccentricity


Exercises 9–12 give the foci or vertices and the eccentricities of ellipses centered at the origin of the xy-plane. In each case, find the ellipse’s standard-form equation in Cartesian coordinates.


Vertices: (±10,0)

Eccentricity: 0.24

Textbook Question

Tangent Lines to Parametrized Curves


In Exercises 1−14, find an equation for the line tangent to the curve at the point defined by the given value of t. Also, find the value of d²y/dx² at this point.


x = sec² t − 1, y = tan t, t = −π/4

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Textbook Question

Polar to Cartesian Equations


Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.


r = 3 cos θ

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