Trigonometry (11th Edition)
Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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finish the chart and graph it please

### Parametric Equations and Elimination of Parameters #### Given Parametric Equations: - \( x = 3 \cos t - 3 \) - \( y = 3 \sin t + 1 \) - Interval: \( 0 \leq t \leq 2\pi \) #### Table of Values: | \( t \) | \( x \) Calculation | \( y \) Calculation | \( (x, y) \) | |-----------|----------------------------------|--------------------------------|--------------| | 0 | \( 3 \cos(0) - 3 = 0 \) | \( 3 \sin(0) + 1 = 1 \) | \( (0, 1) \) | | \(\pi/4\) | \( 3 \cos(\pi/4) - 3 = 3\sqrt{2}/2 - 3 \) | \( 3 \sin(\pi/4) + 1 = 3\sqrt{2}/2 + 1 \) | | | \(\pi/2\) | | | | | \(3\pi/4\)| | | | | \(\pi\) | | | | | \(5\pi/4\)| | | | | \(3\pi/2\)| | | | | \(7\pi/4\)| | | | | \(2\pi\) | | | | #### Steps to Eliminate the Parameter: To eliminate the parameter, express \( x \) and \( y \) in terms of each other without \( t \). #### Graph Explanation: The graph on the right is a coordinate plane labeled with the x-axis and y-axis. Several points derived from the table above are plotted to visualize the parametric equations' path. Each plotted point corresponds to a calculated \( (x, y) \) pair from the table. The path depicted traces out a circle due to the trigonometric sine and cosine functions with amplitudes from \( t = 0 \) to \( t = 2\pi \). Through this activity, learners will understand how parametric equations define a path in the coordinate plane and explore the relationship between trigonometric functions and circular movement.
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Transcribed Image Text:### Parametric Equations and Elimination of Parameters #### Given Parametric Equations: - \( x = 3 \cos t - 3 \) - \( y = 3 \sin t + 1 \) - Interval: \( 0 \leq t \leq 2\pi \) #### Table of Values: | \( t \) | \( x \) Calculation | \( y \) Calculation | \( (x, y) \) | |-----------|----------------------------------|--------------------------------|--------------| | 0 | \( 3 \cos(0) - 3 = 0 \) | \( 3 \sin(0) + 1 = 1 \) | \( (0, 1) \) | | \(\pi/4\) | \( 3 \cos(\pi/4) - 3 = 3\sqrt{2}/2 - 3 \) | \( 3 \sin(\pi/4) + 1 = 3\sqrt{2}/2 + 1 \) | | | \(\pi/2\) | | | | | \(3\pi/4\)| | | | | \(\pi\) | | | | | \(5\pi/4\)| | | | | \(3\pi/2\)| | | | | \(7\pi/4\)| | | | | \(2\pi\) | | | | #### Steps to Eliminate the Parameter: To eliminate the parameter, express \( x \) and \( y \) in terms of each other without \( t \). #### Graph Explanation: The graph on the right is a coordinate plane labeled with the x-axis and y-axis. Several points derived from the table above are plotted to visualize the parametric equations' path. Each plotted point corresponds to a calculated \( (x, y) \) pair from the table. The path depicted traces out a circle due to the trigonometric sine and cosine functions with amplitudes from \( t = 0 \) to \( t = 2\pi \). Through this activity, learners will understand how parametric equations define a path in the coordinate plane and explore the relationship between trigonometric functions and circular movement.
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