Problem 112 - Precalculus with Limits

Question

In Exercises 109-118, describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. \(\theta=3\pi/4\)

Step-by-Step Solution

Answer

The graph of the polar equation \(\theta=3\pi/4\) is the vertical line \(x = -1/\sqrt{2}\) in the rectangular coordinate system.

Step 1: Definition and Conversion

In a polar system, a point in a plain is described by two numbers \(\theta\) and \(r\). Here, \(\theta\) represents the angle and \(r\) represents the distance from a reference point (pole). In this case, we have \(\theta=3\pi/4\) and \(r\) can be any real number. To convert to rectangular coordinates, we would normally use the transformation equations \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). In this case, as \(\theta\) is constant, not dependent on \(r\), the graph in rectangular coordinates would be a vertical line. To find the \(x\) coordinate of the line, we plug \(\theta\) into the \(x\) transforming equation. This gives us \(x = r \cos(3\pi/4)\), which simplifies to \(x = -r /\sqrt{2}\). As \(r\) can be any real number, \(x\) is just a constant, resulting \(x = -1/\sqrt{2}\).

Step 2: Sketching

The corresponding rectangular equation is \(x = -1/\sqrt{2}\). It's a vertical line on the plane. To sketch it, simply draw a line through the point \((-1/\sqrt{2}, y)\), where \(y\) can be any real number.

Key Concepts

Understanding Rectangular Equations

Rectangular equations are expressions that define a curve or line using the Cartesian coordinate system. In this system, every point on the plane is described by an

• X-coordinate: Represents the horizontal position
• Y-coordinate: Represents the vertical position

This type of equation typically describes familiar shapes like lines, circles, and ellipses. For example, a vertical line can be easily represented as a constant equation such as \( x = \text{constant} \). In our exercise, the rectangular equation derived from the polar equation \( \theta=3\pi/4 \) is \( x = -1/\sqrt{2} \), indicating a vertical line. It's crucial to understand that this format is very intuitive when visualizing the position of lines and curves on a graph.
By expressing the curve in rectangular form, it becomes much easier to sketch it on a standard grid.

Graph Conversion Explained

Graph conversion is the process of transforming an equation from one coordinate system to another. Here, we dealt with converting a polar equation to a rectangular one. This is particularly useful when you want to take advantage of the strengths of both coordinate systems.
To perform this conversion, we use the formulas:

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

These formulas allow you to translate information from the polar form, typically used for circles or spirals, to the rectangular form that's accustomed for linear graphs.
In this exercise, the angle \( \theta \) was kept constant at \( 3\pi/4 \), which simplified the conversion process since \( r \) becomes arbitrary, making \( x \) a constant value. This means the equation resolved into the simple linear form \( x = -1/\sqrt{2} \), now expressed clearly in a rectangular grid format.
Such transformations help visualize problems from different perspectives, enhancing overall understanding.

Basics of Polar Coordinates

Polar coordinates present a unique system in which points on a plane are determined by a distance and an angle, rather than the straight Cartesian grid we're used to.
With polar coordinates:

• \( r \): Denotes the distance from the origin to the point
• \( \theta \): Expresses the angle formed with respect to the positive x-axis

This system particularly shines when dealing with problems involving circles and angles, as it directly taps into the relationship between distance and rotation.
In our exercise, the equation \( \theta = 3\pi/4 \) represents all the points that form this angle with the x-axis, which, surprisingly, translates into a line in rectangular coordinates, not an arc. Thus, although polar coordinates might initially seem unconventional, they can simplify the representation of certain types of mathematical relationships, making them an essential tool in advanced math.