a) x 3cost-3, y3sint+1,0si<2a x = 3 (ost-3 y = 3smt+| 3 (os (6)-3-0 351n(0)+/= (x,y) 8TY T/4 T/2 Зп/4 3cos()-3 =0 3 sin)+1=3. Co.9,3.) 4. X. 5T/4 Зп/2 7n/4 8 -4 -2 2. 4. 6. 8 -2 2n -6 Steps for Eliminate the Parameter

Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.
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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.
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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