Problem 57 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$\theta=\frac{\pi}{4}$$

Step-by-Step Solution

Answer

The rectangular form of the given polar equation \(\theta = \frac{\pi}{4}\) is \(x^2 = y^2\).

Step 1: Convert the Polar Coordinate

We have the polar coordinate \( \theta = \frac{\pi}{4} \). To convert an equation in polar coordinates to rectangular coordinates, we use the equations \(x = r \cos \theta\) and \(y = r \sin \theta\). But here, we want to express \(x\) and \(y\) in terms of \(\theta\) alone. To bring into the rectangular form, isolate \(r\) or represent \(r\) in terms of \(x\) and \(y\) using the relationship \(r = \sqrt{x^2 + y^2}\), and then substitute this into each of the equations.

Step 2: Represent in terms of x and y

If we substitute \(\theta = \frac{\pi}{4}\) into the equations, we get \(x = r \cos \frac{\pi}{4}\) and \(y = r \sin \frac{\pi}{4}\). As \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\), this gives us \(x = \frac{r}{\sqrt{2}}\) and \(y = \frac{r}{\sqrt{2}}\), which can also be written as \( r = x \sqrt{2} = y \sqrt{2}\).

Step 3: Solve for the Unknown

As we want to get rid of \(r\), we need to substitute \(r\) from one equation to another. To do this, set \(x \sqrt{2} = y \sqrt{2}\). Squaring both sides we achieve \(x^2 = y^2\).

Key Concepts

Polar Coordinates

When describing the position of a point, polar coordinates offer an alternative to the familiar Cartesian, or rectangular, coordinate system. In polar coordinates, each point is determined by two values: the radial distance from the origin, denoted as \(r\), and the angle \(\theta\) formed with the positive x-axis.

Polar coordinates are particularly useful in cases where the nature of a problem is radial, such as in problems involving circular motion. Instead of specifying a point's location by its x and y coordinates, you describe how far away and at what angle the point is from a central point, called the pole (analogous to the origin in rectangular coordinates).

To recap, polar coordinates consist of:\

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• \(r\): The radial distance from the origin.\
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• \(\theta\): The angle from the positive x-axis to the point, measured in radians or degrees.\
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Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common coordinate system used to represent points in a plane. Each point is defined by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the y-axis and \(y\) is the vertical distance from the x-axis.

The rectangular coordinate system is highly useful for visualizing and solving problems in a straightforward, grid-like fashion, making it easy to calculate distances, slopes, and areas. Algebraically, it's preferred for problems that involve linear relationships and functions.

Essential components of rectangular coordinates include:\

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• \(x\)-coordinate: The horizontal position relative to the vertical (y) axis.\
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• \(y\)-coordinate: The vertical position relative to the horizontal (x) axis.\
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Polar to Rectangular Conversion

Converting polar equations to rectangular form involves translating the radial and angular information into horizontal and vertical coordinates. This translation is based on trigonometric relationships and the Pythagorean theorem. The conversion is achieved using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\).

When we have a polar equation such as \(\theta = \frac{\pi}{4}\), it means that for any point on the line at this angle, the relationship between the x and y coordinates follows a specific pattern. Since \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}\), the polar equation can be expressed in rectangular form as \(x^2 = y^2\). This means that for any point on the line, the distance of the point from the x-axis is equal to its distance from the y-axis, indicating that the line is at a 45-degree angle to both axes and passes through the origin.

To improve understanding of the polar to rectangular conversion process, consider visualizing the unit circle and the association between the angle \(\theta\) and the corresponding coordinates on the circle. It may also be helpful to demonstrate this conversion with different values of \(r\) and \(\theta\), to show how a given polar equation translates into various points in rectangular form.