Problem 71 - Precalculus with Limits

Question

In Exercises 65-84, convert the rectangular equation to polar form. Assume \(a>0\). \(y=-2\)

Step-by-Step Solution

Answer

The polar form of the given rectangular equation \(y=-2\) is \(r\sin(\theta)=-2\).

Step 1: Understanding the Given Equation

Given the rectangular equation \(y=-2\). Here, \(y\) is constant which implies the line is horizontal.

Step 2: Converting to Polar Form

The main equation to be used in converting from rectangular to polar coordinates is \(y=r\sin(\theta)\). Substitute \(r\sin(\theta)\) in place of \(y\) in the given equation to get: \(r\sin(\theta)=-2\).

Step 3: Simplifying the Equation

In this case, \(r\sin(\theta)=-2\) is already in the simplest form. Therefore, it is unnecessary to simplify it further.

Key Concepts

Rectangular to Polar Conversion

Rectangular to Polar Conversion involves changing from a coordinate system defined by x and y (rectangular) to one defined by r and \( \theta \) (polar). This process is crucial when dealing with problems that are more easily expressed or solved in polar coordinates.

To convert from rectangular coordinates to polar form, you use the relationships:

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

In the given exercise, you're converting the equation \( y = -2 \) from rectangular to polar form.
This involves substituting the polar equivalent, \( r\sin(\theta) \), into the equation instead of \( y \). Thus, \( r\sin(\theta) = -2 \) becomes the polar form of the given rectangular equation.

This conversion is straightforward since the equation is linear and much easier when \( y \) represents a constant value.

Trigonometric Equations

Trigonometric Equations involve using trigonometric identities to solve for angles and radii in polar coordinates. In the conversion process from rectangular to polar coordinates, an equation occasionally involves basic trigonometric functions like \( \sin \), \( \cos \), and \( \tan \).

In this context, to solve for \( r \) or \( \theta \), understanding basic trigonometric identities and equations becomes vital. For example, in the polar form \( r\sin(\theta) = -2 \), we essentially apply the trigonometric identity of sine (\( \sin(\theta) \)), demonstrating its relation to the radius and angle.

These equations are simplified directly (if possible) or solved using algebraic manipulations and identities:

• Knowing \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Being able to express one function in terms of another: \( \cos(\theta) = \sqrt{1 - \sin^2(\theta)} \)

Such knowledge can allow us to manipulate the equations into a solvable form and convert complex expressions smoothly.

Coordinate Systems

Coordinate Systems are frameworks that allow us to represent points, lines, and curves within a space. Rectangular and polar are two primary types of coordinate systems.

Rectangular, or Cartesian coordinates, use a grid system to define points with an (x, y) pair. This setup is most useful in scenarios with linear relationships or when dealing with perpendicular bisectors.

Polar coordinates, on the other hand, define a point by its distance from a central point (origin) and the angle from a fixed direction (typically the positive x-axis). This system is often applied in physics and engineering, allowing for simpler representations of circular or spiral structures.

When converting from a rectangular equation like \( y = -2 \) to polar form, understanding both systems is fundamental. You see how linear equations like horizontal or vertical lines in rectangular coordinates become simple trigonometric expressions in polar form. This process can highlight the power of different coordinate systems, showing how each has specific advantages depending on the problem context.

Mastery of both systems leads to versatile problem-solving techniques, giving you the flexibility to choose the most efficient method for different scenarios.