Problem 45 - Precalculus: Functions and Graphs

Question

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. $$\theta=\pi / 4$$

Step-by-Step Solution

Answer

In Cartesian coordinates, the equation is \( y = x \). The graph in the polar plane is a line at \( \theta = \frac{\pi}{4} \).

Step 1: Convert Polar Equation to Cartesian Equation

In polar coordinates, we have the equation \(\theta = \frac{\pi}{4}\). This represents lines in the form \(\theta = \text{constant}\), which can be converted to Cartesian coordinates using the relations: \(x = r\cos\theta\) and \(y = r\sin\theta\). Since \(\tan\theta = \frac{y}{x}\), for \(\theta = \frac{\pi}{4}\), we get \(\tan\frac{\pi}{4} = 1\), hence \(y = x\).

Step 2: Sketch the Graph in the Polar Plane

The polar equation \(\theta = \frac{\pi}{4}\) describes a straight line through the origin making an angle of \(\frac{\pi}{4}\) radians with the positive x-axis. In the polar plane, this would appear as a line extending from the origin at such an angle.

Key Concepts

Polar coordinates

Polar coordinates are a two-dimensional coordinate system where each point on a plane is determined by a distance and an angle. Instead of using standard coordinates like

• x (horizontal position)
• y (vertical position)

Polar coordinates use:

• r (the distance from the origin, or radial coordinate)
• \( \theta \) (the angle measured from the positive x-axis, or angular coordinate)

This system is especially useful for graphing circles, spirals, and other curves that appear naturally in polar form. The main advantage of polar coordinates is that they simplify the representation of some curves and can make calculations easier. However, converting from polar to more familiar Cartesian coordinates often requires some additional steps.

Cartesian coordinates

Cartesian coordinates are a standard system used in mathematics to locate points on a plane. They rely on two numbers that represent a point's location:

• The horizontal distance from the origin, noted as \( x \)
• The vertical distance from the origin, noted as \( y \)

This coordinate system is often used because it is straightforward and aligns with basic arithmetic operations. However, for certain problems, such as those involving circles or periodic curves, Cartesian coordinates can be harder to use compared to polar coordinates. Luckily, there is a way to transition between these two systems using conversion formulas:

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

These relationships allow us to convert from polar coordinates to Cartesian coordinates and vice versa, making it easier to understand equations like \( \theta = \pi /4 \).

Graphing polar equations

When graphing polar equations, the goal is to represent them in a way that visually shows their relationship between \( r \) and \( \theta \). Each line or curve expressed in a polar equation has a unique representation in the polar plane. To graph a simple polar equation like \( \theta = \pi /4 \):

• Understand that this equation results in a line that forms a specific angle with the positive x-axis, which is 45 degrees in this case (because \( \pi /4 \) radians is equivalent to 45 degrees).
• Begin drawing this line at the origin, and extend it outwards at a 45-degree angle.

This type of line will extend infinitely in the direction defined by the angle \( \theta \), and it can be easily visualized in the polar plane. This understanding is crucial for converting and recognizing graphs from polar to Cartesian forms, like turning \( \theta = \pi /4 \) into \( y = x \).

Linear equations

Linear equations are algebraic equations that graph as straight lines on a coordinate plane. The simplest form of a linear equation in Cartesian coordinates is \( y = mx + b \), where:

• \( m \) is the slope of the line, describing its steepness and direction.
• \( b \) represents the y-intercept, where the line crosses the y-axis.

For an equation resulting from the conversion of polar to Cartesian, such as \( y = x \), the slope \( m = 1 \), indicating a 45-degree angle line through the origin (since the x and y values increase at the same rate). Such equations are foundational in algebra, providing a basis for understanding more complex mathematical concepts. In this context, converting \( \theta = \pi /4 \) to \( y = x \) creates a clear visualization of how a polar angle translates into a linear graph.