Problem 62
Question
Find a rectangular equation that has the same graph as the given polar equation. $$ r=3+3 \sec \theta $$
Step-by-Step Solution
Verified Answer
The rectangular equation is \( x^4 + x^2y^2 - 6x^3 - 6xy^2 + 9y^2 = 0 \).
1Step 1: Convert Polar to Cartesian Coordinates Step 1: Use Polar to Rectangular Translations
To begin, recall the basic relationships between polar and rectangular coordinates:\[ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta). \]Additionally, the polar equation includes \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Therefore, rewrite the polar equation:\[ r = 3 + \frac{3}{\cos(\theta)} = 3 + 3\sec(\theta). \]
2Step 2: Convert Polar to Cartesian Coordinates Step 2: Isolate Terms and Simplify
Multiply both sides of the equation by \( \cos(\theta) \) to eliminate the secant:\[ r\cos(\theta) = 3\cos(\theta) + 3. \]Since \( x = r\cos(\theta) \), replace \( r\cos(\theta) \) with \( x \):\[ x = 3\cos(\theta) + 3. \]
3Step 3: Convert Polar to Cartesian Coordinates Step 3: Replace Trigonometric Functions
Substitute \( \cos(\theta) = \frac{x}{r} \):\[ x = 3\left(\frac{x}{r}\right) + 3. \]Multiply through by \( r \) to clear the fraction:\[ xr = 3x + 3r. \]
4Step 4: Convert Polar to Cartesian Coordinates Step 4: Substitute r with Rectangular Components
Since \( r^2 = x^2 + y^2 \), then \( r = \sqrt{x^2 + y^2} \). Substitute \( r \) back in:\[ x\sqrt{x^2 + y^2} = 3x + 3\sqrt{x^2 + y^2}. \]
5Step 5: Solve the Rectangular Equation Step 5: Isolate and Square Both Sides
Rearrange to isolate the square root:\[ x\sqrt{x^2 + y^2} - 3\sqrt{x^2 + y^2} = 3x. \]Factor out \( \sqrt{x^2 + y^2} \):\[ (x - 3)\sqrt{x^2 + y^2} = 3x. \]Square both sides to eliminate the square root:\[ (x - 3)^2(x^2 + y^2) = 9x^2. \]
6Step 6: Final Simplification and Expression Step 6: Expand and Simplify
Expand both sides to obtain the rectangular equation:\[(x^2 - 6x + 9)(x^2 + y^2) = 9x^2.\]Distribute to get:\[x^4 + x^2y^2 - 6x^3 - 6xy^2 + 9x^2 + 9y^2 = 9x^2.\]Move terms to one side to present a proper equation:\[x^4 + x^2y^2 - 6x^3 - 6xy^2 + 9y^2 = 0.\]
Key Concepts
Polar CoordinatesRectangular CoordinatesTrigonometric Identities
Polar Coordinates
Polar coordinates provide an alternative way to represent points in a plane, using an angle and a distance. Typically, the point is described by \((r, \theta)\).
Here, \(r\) is the distance from the origin to the point, while \(\theta\) is the angle between the positive x-axis and a line connecting the origin to the point. The angle is usually measured in radians.
Polar coordinates are particularly useful when dealing with curves and shapes that are circular or spiral in nature. Unlike rectangular (or Cartesian) coordinates which use \((x, y)\) to specify a location, polar coordinates can provide a more intuitive description for points involving rotations or radial distances.
This system is efficient for cases where radial symmetry is involved, such as in physics for describing planetary orbits or electromagnetic waves, as well as in various engineering fields.
Here, \(r\) is the distance from the origin to the point, while \(\theta\) is the angle between the positive x-axis and a line connecting the origin to the point. The angle is usually measured in radians.
Polar coordinates are particularly useful when dealing with curves and shapes that are circular or spiral in nature. Unlike rectangular (or Cartesian) coordinates which use \((x, y)\) to specify a location, polar coordinates can provide a more intuitive description for points involving rotations or radial distances.
This system is efficient for cases where radial symmetry is involved, such as in physics for describing planetary orbits or electromagnetic waves, as well as in various engineering fields.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, use pairs of numbers \((x, y)\) to define positions on a plane. Each point has a unique pair which specifies its exact location based on its distance from two perpendicular axes.
The coordinate \(x\) represents the point's horizontal position, and \(y\) represents its vertical position. This coordinate system is straightforward and widely used because of its simplicity and ease of visualization on a standard grid or graph.
The coordinate \(x\) represents the point's horizontal position, and \(y\) represents its vertical position. This coordinate system is straightforward and widely used because of its simplicity and ease of visualization on a standard grid or graph.
- Rectangular coordinates can easily represent linear relationships, alignments with axes, and facilitate calculations in algebra or calculus.
- They also allow a seamless translation into a computer graphical display, such as plots or charts, which is essential in various fields like data science, architecture, and computer graphics.
Trigonometric Identities
Trigonometric identities are mathematical equations that involve trigonometric functions like sine, cosine, and tangent. These functions relate the angles of a triangle to the lengths of its sides in right-angle triangles.
However, they are also crucial when switching between polar and rectangular coordinates. In the conversion process, recognizing identities such as \( \sec(\theta) = \frac{1}{\cos(\theta)} \) becomes essential. This helps in rewriting and simplifying polar equations into equivalents in rectangular coordinates.
For the given exercise, understanding that \( \sec(\theta) \) can be rewritten as \( \frac{1}{\cos(\theta)} \) allows us to manipulate the polar equation and ultimately translate it into a rectangular equation.
However, they are also crucial when switching between polar and rectangular coordinates. In the conversion process, recognizing identities such as \( \sec(\theta) = \frac{1}{\cos(\theta)} \) becomes essential. This helps in rewriting and simplifying polar equations into equivalents in rectangular coordinates.
For the given exercise, understanding that \( \sec(\theta) \) can be rewritten as \( \frac{1}{\cos(\theta)} \) allows us to manipulate the polar equation and ultimately translate it into a rectangular equation.
Other exercises in this chapter
Problem 60
Find a rectangular equation that has the same graph as the given polar equation. $$ r(4-\sin \theta)=10 $$
View solution Problem 61
Find a rectangular equation that has the same graph as the given polar equation. $$ r=\frac{5}{3 \cos \theta+8 \sin \theta} $$
View solution Problem 65
In rectangular coordinates the \(x\) -intercepts of the graph of a function \(y=f(x)\) are determined from the solutions of the equation \(f(x)=0 .\) In the nex
View solution Problem 59
Find a rectangular equation that has the same graph as the given polar equation. $$ r=\frac{2}{1+3 \cos \theta} $$
View solution