Problem 65 - Precalculus Mathematics for Calculus

Question

Convert the polar equation to rectangular coordinates. $$r^{2}=\tan \theta$$

Step-by-Step Solution

Answer

The rectangular equation is \(x^3 + xy^2 - y = 0\).

Step 1: Understanding the Polar Equation

We are given the polar equation \(r^2 = \tan \theta\) where \(r\) is the radial distance and \(\theta\) is the angular coordinate. Our task is to convert this to a rectangular coordinate system where equations are expressed in terms of \(x\) and \(y\).

Step 2: Identify Known Relationships

Recall the basic relationships between polar and rectangular coordinates: \(x = r \cos \theta\), \(y = r \sin \theta\), \(r^2 = x^2 + y^2\), and the trigonometric identity \(\tan \theta = \frac{y}{x}\). These will help us in converting from polar to rectangular coordinates.

Step 3: Substitute the Trigonometric Identity

Substitute \(\tan \theta = \frac{y}{x}\) into the given polar equation. This yields: \[r^2 = \tan \theta = \frac{y}{x}\] Thus, \[r^2 = \frac{y}{x}\].

Step 4: Replace \(r^2\) with Rectangular Coordinates

Since \(r^2 = x^2 + y^2\), we substitute this into the equation: \[x^2 + y^2 = \frac{y}{x}\].

Step 5: Eliminate Fractions

Multiply both sides of the equation by \(x\) to eliminate the fraction: \[x(x^2 + y^2) = y\].

Step 6: Simplify the Equation

Distribute \(x\) on the left side of the equation: \[x^3 + xy^2 = y\].

Step 7: Formulate the Rectangular Equation

Rearrange the equation to have all terms on one side: \[x^3 + xy^2 - y = 0\]. This is the equation in rectangular coordinates.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a standard system of coordinates used widely in mathematics and engineering. They are defined by an ordered pair

• ( x, y ),
• where x represents the horizontal position,
• and y represents the vertical position.

This system is excellent for graphing algebraic equations and visualizing points in a two-dimensional plane.
The axes in this system intersect at the origin, point (0, 0), which is the cornerstone for defining any other point's position.
Rectangular coordinates are particularly useful because:

• The distance between two points can be easily calculated using thePythagorean theorem,
• The transformation from polar coordinates allows for the convenient conversion of equations and problem statements into familiar algebraic expressions.

Understanding rectangular coordinates helps us interpret complex data clearly, making it invaluable for solving mathematical problems.

Polar Coordinates

Polar coordinates provide a different approach to specifying the location of a point in a plane compared to the more familiar rectangular coordinates.
In this system, each point is defined by:

• the radial distance ( r ) from a fixed point called the pole (like the origin in Cartesian coordinates),
• and an angle ( \( heta \) ) measured from a fixed direction, typically the positive x-axis.

This approach is particularly advantageous in scenarios involving circular and spiral patterns, where angles and radii are more intuitive than horizontal and vertical measurements.
Polar coordinates are complementary to rectangular coordinates, and understanding both is crucial asthey offer flexibility in analyzing different types of problems in geometry and physics.
They provide unique perspectives:

• Enhancing strategies for evaluating trigonometric equations through visualization,
• And solving problems involving rotational symmetry more efficiently.

By mastering both polar and rectangular coordinates, you can transform mathematical expressions seamlessly between forms to suit specific needs.

Trigonometric Identity

A trigonometric identity is an equation that is true for all values of the variable, usually involving trigonometric functions.
These identities are essential tools in simplifying expressions or solving equations that include angles and lengths in a triangle.
One key identity used in polar to rectangular conversions is

• \( \tan \theta = \frac{y}{x} \),

which relates the angle \( \theta \) to the y and x coordinates in rectangular terms.
Other foundational identities involve:

• Sine ( \( \sin \theta = \frac{opp}{hyp} \) ),
• Cosine ( \( \cos \theta = \frac{adj}{hyp} \) ),
• and the Pythagorean identity ( \( \sin^2 \theta + \cos^2 \theta = 1 \) ).

These identities are not only used in geometry but also play a crucial role in calculus and physics, where they assist in calculating derivatives and integrating functions.
By understanding these trigonometric relationships, we can skillfully switch between different systems of equations, aiding in the seamless transformation between polar and rectangular systems.

Coordinate Transformation

Coordinate transformation is the process of converting the expression of data from one coordinate system to another.
In our exercise, we converted polar coordinates to rectangular coordinates, showcasing how transformations simplify the mathematical representation of a problem.
The common transformations include:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \tan^{-1}(\frac{y}{x}) \)

These equations enable the substitution of polar terms with rectangular ones, facilitating easier manipulation of the given problem or equation.
Using these transformations, we are able to convert the polar equation \(r^2 = \tan \theta\) into its equivalent rectangular form \(x^3 + xy^2 - y = 0\), allowing us to analyze it using more traditional algebraic methods.
This flexibility aids in problem-solving across disciplines, from physics to engineering, and this underscores the importance of mastering both coordinate systems.
Each transformation provides a valuable method of tailoring equations and enhancing understanding based on the problem's context.