Trigonometry (MindTap Course List)
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.
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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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Trigonometry (MindTap Course List)
Trigonometry
ISBN:9781305652224
Author:Charles P. McKeague, Mark D. Turner
Publisher:Cengage Learning