Problem 51 - Contemporary Precalculus

Question

Find a rectangular equation that is equivalent to the given polar equation. $$r=\frac{4}{1+\sin \theta}$$

Step-by-Step Solution

Answer

Question: Find the rectangular equation equivalent to the polar equation r = 4/(1 + sin θ). Answer: The rectangular equation equivalent to the given polar equation is x² - y² + 8y - 16 = 0.

Step 1: Identify relationships

First, recall the following relationships: 1. $$x = r\cos{\theta}$$ 2. $$y = r\sin{\theta}$$ 3. $$r = \sqrt{x^2 + y^2}$$ We'll make use of these relationships to turn the polar equation into a rectangular equation.

Step 2: Substitute r*sinθ for y

Using the second relationship, we know that $$y = r\sin{\theta}$$ Now, rewriting the equation r = 4/(1 + sin θ), we have, $$r=\frac{4}{1+\sin \theta}$$ So we will first try to eliminate sin θ from the polar equation. We do that by multiplying both sides by (1 + sin θ) and get: $$r[1+\sin{\theta}] = 4$$ or $$r+\sin{\theta}r = 4$$ Now using the second relationship from step 1, we can substitute $$r\sin{\theta}$$ for y in this equation: $$r+y=4$$

Step 3: Solve for r

Now, we need to eliminate r in the expression $$r+y=4$$. Using the third relationship, we can express r in terms of x and y: $$r = \sqrt{x^2 + y^2}$$ So replace r in the expression: $$\sqrt{x^2 + y^2}+y=4$$

Step 4: Isolated r term and square both sides

Isolate the r term in the equation $$\sqrt{x^2 + y^2}+y=4$$ by subtracting y from both sides: $$\sqrt{x^2 + y^2} = 4 -y$$ Now, square both sides of the equation to get rid of the square root: $$x^2 + y^2 = (4 - y)^2$$

Step 5: Simplify

Now, expand the right side of the equation and simplify: $$x^2 + y^2 = (4 - y)^2 = 4^2 - 2 \cdot 4 \cdot y + y^2$$ $$x^2 + y^2 = 16 - 8y + y^2$$ Move all the terms to the left side, and we get: $$x^2 + y^2 - 16 + 8y - y^2 = 0$$ $$x^2 - y^2 + 8y - 16 = 0$$ The rectangular equation equivalent to the given polar equation is: $$x^2 - y^2 + 8y - 16 = 0$$

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a two-dimensional plane using the terms \((x, y)\). These coordinates express the position of a point by measuring its distance horizontally from the vertical axis (known as the x-axis) and vertically from the horizontal axis (known as the y-axis).

Rectangular coordinates are named after the French mathematician René Descartes, who introduced the method. They provide a straightforward way to map points in an Euclidean space, making it easier to describe geometric shapes and distances. When dealing with rectangular coordinates:

• \(x\) is the distance along the horizontal axis.
• \(y\) is the distance along the vertical axis.

Understanding rectangular coordinates is crucial in converting them to polar coordinates, especially when you are tasked with transforming equations for different mathematical applications.

Polar Coordinates

Polar coordinates offer a different way of representing points in a plane compared to rectangular coordinates. Instead of using horizontal and vertical distances, polar coordinates use a radius and an angle. This system describes a point's location using \((r, \theta)\), where:

• \(r\) represents the distance from the origin to the point.
• \(\theta\) (theta) is the angle measured from the positive x-axis to the line connecting the origin with the point, usually in radians or degrees.

This method can be particularly useful when dealing with problems exhibiting radial symmetry or circular motion, as polar coordinates naturally adhere to such patterns.

To understand how to convert between polar and rectangular coordinates, it's essential to grasp the basic relationships, such as \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\). These conversions take advantage of trigonometric identities, serving as vital tools in various mathematical contexts.

Mathematical Relationships

Understanding the mathematical relationships between polar and rectangular coordinates is key to transforming equations. To convert from a polar to a rectangular equation, one uses the relationships derived from trigonometry:

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

These formulas create a bridge between the coordinate systems through the interplay of radius, angle, and distances along the axes.

In our example, we first recognized that \(r\sin(\theta) = y\) and, thus, decided to eliminate the \(\sin(\theta)\) component by manipulating the given equation. After making appropriate substitutions, solving for \(r\) using \(r = \sqrt{x^2 + y^2}\) is the next step forward. By knowing these relationships, you can transition fluidly between equation forms.

Equation Transformation

Transforming equations from polar to rectangular form involves utilizing and manipulating the established relationships between the two coordinate systems. In this context, the primary goal is to express the polar equation using \(x\) and \(y\) variables typical of rectangular equations.

• Initially, recognize and substitute corresponding polar expressions based on trigonometric relationships.
• Isolate complex terms, such as \(r\), by using its equivalent in rectangular terms \(\sqrt{x^2 + y^2}\).
• Squaring both sides might be necessary to remove square roots, simplifying the expression further.
• Finally, rearrange and simplify the equation to achieve a clean rectangular form.

In the provided exercise, through a series of algebraic manipulations and substitutions, the polar equation \(r = \frac{4}{1 + \sin \theta}\) is converted into the rectangular form \(x^2 - y^2 + 8y - 16 = 0\). This process illustrates the power and necessity of understanding equation transformation in advanced math problems.