There are 6 points on the curve r= cos(20) where it has horizontal tangents. 2 of these can be found exactly. Find these, and then find the other 4 points (in both polar and rectangular coordinates - find the values of theta, r, x, and y) to 5 decimal places. Please show polar graph
There are 6 points on the curve r= cos(20) where it has horizontal tangents. 2 of these can be found exactly. Find these, and then find the other 4 points (in both polar and rectangular coordinates - find the values of theta, r, x, and y) to 5 decimal places. Please show polar graph
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Finding Points with Horizontal Tangents on the Curve \( r = \cos(2\theta) \)**
The curve described by \( r = \cos(2\theta) \) has six points where it exhibits horizontal tangents. Of these, two points can be determined exactly. Your task is to identify these two exact points, and subsequently, calculate the other four points. This should be done in both polar and rectangular coordinates (to five decimal places). Additionally, please provide a polar graph to illustrate these points.
### Steps to Solve:
1. **Identify Exact Points**: Use trigonometric identities and properties of the cosine function to find the two exact points with horizontal tangents.
2. **Calculate Remaining Points**:
- Convert the polar coordinates to Cartesian coordinates \( (x, y) \).
- Ensure calculations are precise to five decimal places.
3. **Graphical Representation**:
- Create a polar graph of the curve \( r = \cos(2\theta) \).
- Indicate the locations of the points with horizontal tangents.
### Key Concepts:
- **Polar Coordinates**: Relates to expressing points in the form \( (r, \theta) \), where \( r \) represents the radius and \( \theta \) the angle.
- **Rectangular Coordinates**: Converts polar coordinates to \( (x, y) \) using the formulas:
\[
x = r \cos(\theta), \quad y = r \sin(\theta)
\]
- **Horizontal Tangent Criterion**: The derivative of the radial function with respect to the angle should be zero for horizontal tangents.
### Note:
Utilize appropriate software tools or graphing calculators to visualize and verify the solutions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F272afd08-fa1b-4f9b-918c-9c781af320ae%2F7ceb8246-d0a9-454b-8791-3434f32d716b%2Fhfconif_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Points with Horizontal Tangents on the Curve \( r = \cos(2\theta) \)**
The curve described by \( r = \cos(2\theta) \) has six points where it exhibits horizontal tangents. Of these, two points can be determined exactly. Your task is to identify these two exact points, and subsequently, calculate the other four points. This should be done in both polar and rectangular coordinates (to five decimal places). Additionally, please provide a polar graph to illustrate these points.
### Steps to Solve:
1. **Identify Exact Points**: Use trigonometric identities and properties of the cosine function to find the two exact points with horizontal tangents.
2. **Calculate Remaining Points**:
- Convert the polar coordinates to Cartesian coordinates \( (x, y) \).
- Ensure calculations are precise to five decimal places.
3. **Graphical Representation**:
- Create a polar graph of the curve \( r = \cos(2\theta) \).
- Indicate the locations of the points with horizontal tangents.
### Key Concepts:
- **Polar Coordinates**: Relates to expressing points in the form \( (r, \theta) \), where \( r \) represents the radius and \( \theta \) the angle.
- **Rectangular Coordinates**: Converts polar coordinates to \( (x, y) \) using the formulas:
\[
x = r \cos(\theta), \quad y = r \sin(\theta)
\]
- **Horizontal Tangent Criterion**: The derivative of the radial function with respect to the angle should be zero for horizontal tangents.
### Note:
Utilize appropriate software tools or graphing calculators to visualize and verify the solutions.
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