Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r\left(\sin \theta+r \cos ^{2} \theta\right)=1 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( y = 1 - x^2 \), a downward-facing parabola.

Step 1: Polar to Cartesian Conversion

Begin by converting the polar equation into Cartesian coordinates. Recall that the conversions are: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \].The given equation is: \[ r (\sin \theta + r \cos^2 \theta) = 1 \]Substitute \( y = r \sin \theta \) and \( x^2 = (r \cos \theta)^2 \) to rewrite the equation as: \[ r\left( \frac{y}{r} + \frac{x^2}{r} \right) = 1 \]. Simplifying gives: \[ y + x^2 = 1\].

Step 2: Rearration of the Equation

Rearrange the equation \[ y + x^2 = 1 \] to express \( y \) explicitly, yielding: \[ y = 1 - x^2 \]. This is now in Cartesian form: it represents a parabola.

Step 3: Graphing the Equation

Sketch the graph of the equation \( y = 1 - x^2 \) in the Cartesian plane. This is a downward-opening parabola with its vertex at the point \((0, 1)\) and symmetric about the y-axis.

Step 4: Sketching in the Polar Plane

Use the Cartesian graph to assist in sketching the polar graph. The equation in polar form describes similar behavior, equivalent to a parabola opening downward. In the polar plane \((r, \theta)\), points are plotted by first locating \(\theta\) and then moving along the radius \(r\). Use key angles \(\theta\) and calculations from Cartesian \((x, y)\) points to plot in polar form and achieve a similar shape.

Key Concepts

Polar Coordinates

Polar coordinates are a method of representing points in a plane using two values: the radial coordinate, known as \( r \), and the angular coordinate, \( \theta \). These values describe the position of a point:

• \( r \): The distance from the origin to the point in the plane.
• \( \theta \): The angle between the positive x-axis and the line connecting the origin to the point.

Polar coordinates are particularly useful in scenarios where the relationship between variables is circular or spiral in nature. For instance, when dealing with circular or radial symmetry, polar coordinates can simplify mathematical descriptions and solutions.
To convert from polar to Cartesian coordinates, we use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This transformation allows us to represent polar equations in a form compatible with Cartesian graphs.

Cartesian Coordinates

Cartesian coordinates, also known as rectangular coordinates, represent points in a plane using an ordered pair \((x, y)\). This setup is based on two perpendicular lines (axes) that intersect at a point called the origin.

• \( x \): The horizontal distance from the y-axis.
• \( y \): The vertical distance from the x-axis.

The Cartesian coordinate system is widely used in mathematics, engineering, and science for its simplicity in representing straight-line distances and intersections. It allows for straightforward plotting of linear relationships and other shapes such as parabolas, circles, and ellipses.
In this exercise, converting from polar to Cartesian form revealed an equation \( y = 1 - x^2 \), which is easier to interpret and graph on a Cartesian plane. This conversion process is crucial in making complex polar equations more tractable for analysis and graphing.

Parabola

A parabola is a U-shaped curve that can open either up, down, left, or right depending on its equation. In the Cartesian plane, a parabola is often expressed by the equation \( y = ax^2 + bx + c \). In this version, \( y = 1 - x^2 \), the parabola:

• Opens downward.
• Has its vertex at the point \((0, 1)\).
• Is symmetric about the y-axis.

Parabolas have unique characteristics such as the vertex, focus, and directrix that define their shape and position. The vertex is the highest or lowest point on the parabola, and it acts as the point of symmetry.
To graph the parabola \( y = 1 - x^2 \), we start at the vertex at \( (0, 1) \) and plot points from the vertex that lie equidistant from the y-axis. The graph then curves downward in a symmetrical manner. Understanding parabolas is fundamental in diverse fields such as physics, engineering, and economics due to their natural occurrence and simple yet powerful mathematical properties.