150 135 120° 105° 90° 180° 165* 240 225 210 .592 .042 75° .582 60° .00€ 45° 315* 30° 330⁰ 15° 8=0° Suppose you graph the function r(0) = 4cos (20) on the interval [0,2π] in polar coordinates. How many times do you think the graph will pass through the origin on this interval? How many distinct "loops" will the graph make? See if you can anticipate the answers to these questions either using algebra or graphing by hand. Then, use a graphing calculator or Desmos to verify your guess.
150 135 120° 105° 90° 180° 165* 240 225 210 .592 .042 75° .582 60° .00€ 45° 315* 30° 330⁰ 15° 8=0° Suppose you graph the function r(0) = 4cos (20) on the interval [0,2π] in polar coordinates. How many times do you think the graph will pass through the origin on this interval? How many distinct "loops" will the graph make? See if you can anticipate the answers to these questions either using algebra or graphing by hand. Then, use a graphing calculator or Desmos to verify your guess.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.5: Polar Coordinates
Problem 14E
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8
![**Educational Content: Graphing in Polar Coordinates**
### Polar Graph Explanation
Above is a polar graph with concentric circles and radial lines marked in degrees from 0° to 360°. The circles represent constant radius values from the center, while the radial lines represent angles. This graph serves as a framework for plotting polar equations, which express points in terms of a radius \( r \) and an angle \( \theta \).
### Exercises
8. **Graph the Function**:
- Consider the function \( r(\theta) = 4 \cos (2\theta) \) on the interval \([0, 2\pi]\).
- **Questions to Explore**:
- How many times does the graph pass through the origin within this interval?
- How many distinct "loops" does the graph make?
- **Tasks**:
- Anticipate the answers using algebraic techniques or by sketching the graph manually.
- Confirm your predictions using a graphing calculator or software like Desmos.
9. **New Function Evaluation**:
- Follow the same procedures for graphing and questioning, using the function \( r(\theta) = 4 \cos (3\theta) \).
10. **Function Representation**:
- **Understanding Functions**:
- **(a)** In rectangular coordinates, determine if a graph represents a function.
- **(b)** Investigate how to discern function representation in polar coordinates.
- **(c)** Use a graphing tool (e.g., Desmos) for \( r(\theta) = 3 \cos (2.1\theta) \). Evaluate whether this graph represents a function and justify your conclusions.
These exercises are designed to strengthen your understanding of graphing in polar coordinates and analyzing the resulting plots for function characteristics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e869531-958e-4f8b-997d-b908634d80a9%2F20f19105-da44-4534-85a9-766f30cd736e%2Fe24fod_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Content: Graphing in Polar Coordinates**
### Polar Graph Explanation
Above is a polar graph with concentric circles and radial lines marked in degrees from 0° to 360°. The circles represent constant radius values from the center, while the radial lines represent angles. This graph serves as a framework for plotting polar equations, which express points in terms of a radius \( r \) and an angle \( \theta \).
### Exercises
8. **Graph the Function**:
- Consider the function \( r(\theta) = 4 \cos (2\theta) \) on the interval \([0, 2\pi]\).
- **Questions to Explore**:
- How many times does the graph pass through the origin within this interval?
- How many distinct "loops" does the graph make?
- **Tasks**:
- Anticipate the answers using algebraic techniques or by sketching the graph manually.
- Confirm your predictions using a graphing calculator or software like Desmos.
9. **New Function Evaluation**:
- Follow the same procedures for graphing and questioning, using the function \( r(\theta) = 4 \cos (3\theta) \).
10. **Function Representation**:
- **Understanding Functions**:
- **(a)** In rectangular coordinates, determine if a graph represents a function.
- **(b)** Investigate how to discern function representation in polar coordinates.
- **(c)** Use a graphing tool (e.g., Desmos) for \( r(\theta) = 3 \cos (2.1\theta) \). Evaluate whether this graph represents a function and justify your conclusions.
These exercises are designed to strengthen your understanding of graphing in polar coordinates and analyzing the resulting plots for function characteristics.
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