Problem 88 - Precalculus with Limits

Question

In Exercises 85-108, convert the polar equation to rectangular form. \(r=-5\ \sin\ \theta\)

Step-by-Step Solution

Answer

After all transformations and substitutions, the final form of the rectangular equation derived from the polar equation \(r=-5\ \sin\ \theta\) is \(y = -5 +5x^2/r^2\), where \(r = \sqrt{x^2 + y^2}\).

Step 1: Identify the polar equation

The given polar equation is \(r=-5\ \sin\ \theta\). We need to convert this equation into its rectangular form.

Step 2: Use the relationship between polar and rectangular coordinates

We know that in polar coordinates, \(y = r \sin(\theta)\). We can substitute \(r=-5\ \sin\ \theta\) into this equation to give: \(y=(-5\ \sin\theta) \sin\ \theta\), therefore \(y = -5 \sin^2\ \theta\).

Step 3: Apply Pythagorean Identity

We know that \(\sin^2\ \theta = 1 - cos^2\ \theta\), we substitute this to the rectangular equation in step 2, it gives: \(y = -5(1 - cos^2\ \theta)\), then distribute the -5: \(y = -5 +5cos^2\ \theta\).

Step 4: Replace \(\cos\ \theta\) with \(x/r\)

We replace \(\cos\ \theta\) with \(x/r\), it gives: \(y = -5 +5(x/r)^2\).

Step 5: Simplify the equation to receive the final form

We simplify the equation to its simplest form: \(y = -5 +5(x^2/r^2)\), further simplifying it gives: \(y = -5 +5x^2/r^2\). Considering r is the radius in polar which is \(r = \sqrt{x^2 + y^2}\), we then substitute r into the equation for final form.

Key Concepts

Polar Coordinates

Polar coordinates provide a different way of representing points in a plane. They use two values: the radial coordinate \(r\) and the angular coordinate \(\theta\).

• The radial coordinate, \(r\), refers to the distance from the origin (center point) to the point in the plane.
• The angular coordinate, \(\theta\), is the angle formed with the positive x-axis.

This system is incredibly helpful in situations where angular relationships are involved, such as in circular or spiral patterns. The polar system simplifies the equations of curves and can be found using circles or any shape that relates around a central point. To convert these coordinates into a more familiar form like rectangular coordinates, we use trigonometric relationships which link angles and distances.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe the location of a point by using a pair \((x, y)\).

• The x-coordinate represents the horizontal position of a point, moving left or right from the origin.
• The y-coordinate represents the vertical position, indicating upward or downward movement from the origin.

This system divides the plane into four quadrants using two axes (x-axis and y-axis), and each point is a result of intersection between the values x and y. Most often used in algebra and geometry, rectangular coordinates are easy to visualize, especially for straightforward shapes like squares and rectangles. When dealing with curves such as circles, polar coordinates can simplify calculations, requiring conversion back to rectangular for most practical applications.

Pythagorean Identity

The Pythagorean identity is a fundamental aspect of trigonometry, closely linked with the concept of right triangles.

• It states that \( \sin^2 \theta + \cos^2 \theta = 1 \).

This identity reflects the inherent relationship between sine and cosine, which correspond to the measures of a triangle's sides in relation to its hypotenuse. Often used in simplifying expressions and solving trigonometric equations, this identity is pivotal when transitioning between polar and rectangular forms. For instance, in our conversion problem, it helps express the polar equation into a form that aligns with rectangular coordinates by substituting \( \sin^2 \theta \), which is part of a rectangle's side.

Coordinate Conversion

Coordinate conversion is the process of transforming a point defined in one coordinate system into another coordinate system. This is essential in mathematical modeling and real-world applications.

• In converting from polar to rectangular coordinates, you use \( x = r \cos \theta \) and \( y = r \sin \theta \).
• For going the other way, use \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1} \left(\frac{y}{x}\right) \).

In our task of converting \( r = -5 \sin \theta \) into rectangular form, understanding these relationships is crucial. It allows us to express the curve in a manner that can be easily arranged and interpreted in a rectangular framework. Coordinate conversion is pivotal in navigation, engineering, and computer graphics, where changing the perspective or form is necessary for easier interpretation or usage.