Trigonometry (MindTap Course List)
8th Edition
ISBN: 9781305652224
Author: Charles P. McKeague, Mark D. Turner
Publisher: Cengage Learning
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![### Problem 2
**Task:**
- Graph \( r = 4 \cos(2\theta) \) without a calculator.
- Then use a calculator to find the length of the arc forming one petal.
**Instructions:**
1. **Graphing \( r = 4 \cos(2\theta) \):**
- Start by understanding that this is a polar equation where \( r \) (radius) is a function of \( \theta \) (angle).
- The equation \( r = 4 \cos(2\theta) \) represents a polar curve known as a "four-leaved rose." It features a periodic, sinusoidal pattern with four symmetrical petals.
- To sketch the graph, calculate the values of \( r \) for various \( \theta \) values across one full period \( [0, 2\pi] \). Note the characteristic symmetry.
2. **Finding the arc length of one petal with a calculator:**
- To calculate the length of the arc forming one petal, use the formula for the arc length \( S \) in polar coordinates:
\[
S = \int_{a}^{b} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 }\, d\theta
\]
- Substitute \( r = 4 \cos(2\theta) \) and find the derivative \( \frac{dr}{d\theta} \).
- Integrate this expression over the limits corresponding to one petal.
This task requires both manual graphing skills and proficiency in calculus to find the arc length accurately.](https://content.bartleby.com/qna-images/question/2bb9483f-e549-4e9c-b819-8ce0593a1865/b745d4cc-b21a-4e34-ab42-a9a0ed1d8e59/q610ct_processed.png)
Transcribed Image Text:### Problem 2
**Task:**
- Graph \( r = 4 \cos(2\theta) \) without a calculator.
- Then use a calculator to find the length of the arc forming one petal.
**Instructions:**
1. **Graphing \( r = 4 \cos(2\theta) \):**
- Start by understanding that this is a polar equation where \( r \) (radius) is a function of \( \theta \) (angle).
- The equation \( r = 4 \cos(2\theta) \) represents a polar curve known as a "four-leaved rose." It features a periodic, sinusoidal pattern with four symmetrical petals.
- To sketch the graph, calculate the values of \( r \) for various \( \theta \) values across one full period \( [0, 2\pi] \). Note the characteristic symmetry.
2. **Finding the arc length of one petal with a calculator:**
- To calculate the length of the arc forming one petal, use the formula for the arc length \( S \) in polar coordinates:
\[
S = \int_{a}^{b} \sqrt{ \left( \frac{dr}{d\theta} \right)^2 + r^2 }\, d\theta
\]
- Substitute \( r = 4 \cos(2\theta) \) and find the derivative \( \frac{dr}{d\theta} \).
- Integrate this expression over the limits corresponding to one petal.
This task requires both manual graphing skills and proficiency in calculus to find the arc length accurately.
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