Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter8: Polar Coordinates And Parametric Equations
Section8.2: Graphs Of Polar Equations
Problem 66E
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The exponential function is a type of mathematical function which is used in real-world contexts. It helps to find out the exponential decay model or exponential growth model, in mathematical models. In this topic, we will understand descriptive rules, concepts, structures, graphs, interpreter series, work formulas, and examples of functions involving exponents.
Question
F: #12.
![## 12. Graph
Given the polar equation:
\[ r = 2 - 2 \sin \theta \]
### Explanation
This equation represents a circle in polar coordinates. To understand the graph, it is essential to know how polar coordinates and their equations translate into visual representations. Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
### Graph Analysis
- **Equation Form**: The given equation is in the form of a circle equation in polar coordinates, where \( r \) is the radius, and \( \theta \) is the angle.
- **Components**:
- \( r \): The radial distance from the origin (pole) to a point on the curve.
- \( \sin \theta \): A trigonometric function that provides the y-coordinate of a point in the unit circle for a given angle \( \theta \).
From the given equation, we can see that it describes a circle shifted downwards in the polar coordinate system. The center of the circle is at (0,-1) in Cartesian coordinates, and the radius is 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c34de7e-5935-4303-bb64-ce025f7422a2%2Fd5c3b43b-4330-4dca-943c-6ea7d5154e8c%2Fzhydxkp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## 12. Graph
Given the polar equation:
\[ r = 2 - 2 \sin \theta \]
### Explanation
This equation represents a circle in polar coordinates. To understand the graph, it is essential to know how polar coordinates and their equations translate into visual representations. Polar coordinates are a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
### Graph Analysis
- **Equation Form**: The given equation is in the form of a circle equation in polar coordinates, where \( r \) is the radius, and \( \theta \) is the angle.
- **Components**:
- \( r \): The radial distance from the origin (pole) to a point on the curve.
- \( \sin \theta \): A trigonometric function that provides the y-coordinate of a point in the unit circle for a given angle \( \theta \).
From the given equation, we can see that it describes a circle shifted downwards in the polar coordinate system. The center of the circle is at (0,-1) in Cartesian coordinates, and the radius is 1.
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