Trigonometry (11th Edition)
11th Edition
ISBN: 9780134217437
Author: Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher: PEARSON
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![### (a) Sketch the Graph of the Parametric Curve
The given parametric equations are:
\[ x = 3 \sin t + 3 \]
\[ y = 2 \cos t \]
\[ \text{where } 0 \leq t \leq \pi \]
The task is to sketch the graph corresponding to these equations.
#### Graphs Overview:
- **Top Left Graph**: Represents the top half-ellipse curve sweeping from left to right.
- **Top Right Graph**: Shows a complete closed ellipse.
- **Bottom Left Graph**: Displays the bottom half-ellipse mirror-inverted from the top.
- **Bottom Right Graph**: Illustrates the top half-ellipse sweeping from center to right.
Choice of the correct graph depends on how these parametric equations transform over the interval \( t \).
### (b) Eliminate the Parameter \( t \)
To find the rectangular form of the given parametric equations, eliminate the parameter \( t \):
Work through the relationships:
\[ x - 3 = 3 \sin t \]
\[ y = 2 \cos t \]
Identity to use:
\[ \sin^2 t + \cos^2 t = 1 \]
Substitute expressions:
\[ \left( \frac{x - 3}{3} \right)^2 + \left( \frac{y}{2} \right)^2 = 1 \]
The curve is an ellipse with the equation for the correct range of \( x \geq 3 \).](https://content.bartleby.com/qna-images/question/aa00ebaa-23b9-4b28-a7ce-1aeaaf3b9d34/22cac988-94d7-434f-965d-eb4a7da7703e/c6dmmk_processed.png)
Transcribed Image Text:### (a) Sketch the Graph of the Parametric Curve
The given parametric equations are:
\[ x = 3 \sin t + 3 \]
\[ y = 2 \cos t \]
\[ \text{where } 0 \leq t \leq \pi \]
The task is to sketch the graph corresponding to these equations.
#### Graphs Overview:
- **Top Left Graph**: Represents the top half-ellipse curve sweeping from left to right.
- **Top Right Graph**: Shows a complete closed ellipse.
- **Bottom Left Graph**: Displays the bottom half-ellipse mirror-inverted from the top.
- **Bottom Right Graph**: Illustrates the top half-ellipse sweeping from center to right.
Choice of the correct graph depends on how these parametric equations transform over the interval \( t \).
### (b) Eliminate the Parameter \( t \)
To find the rectangular form of the given parametric equations, eliminate the parameter \( t \):
Work through the relationships:
\[ x - 3 = 3 \sin t \]
\[ y = 2 \cos t \]
Identity to use:
\[ \sin^2 t + \cos^2 t = 1 \]
Substitute expressions:
\[ \left( \frac{x - 3}{3} \right)^2 + \left( \frac{y}{2} \right)^2 = 1 \]
The curve is an ellipse with the equation for the correct range of \( x \geq 3 \).
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