Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\theta $$

Short answer

The polar equation \( r = \theta \) converts to a parametric form in rectangular coordinates: \( x = \theta \cos(\theta) \), \( y = \theta \sin(\theta) \), describing a spiral graph.

Step-by-step solution

Understand Polar Coordinate Concepts

In polar coordinates, each point in the plane is determined by a distance \( r \) from the origin and an angle \( \theta \) from the positive x-axis. For this problem, the equation given is \( r = \theta \). It describes a spiral where \( r \) increases linearly with \( \theta \). Our task is to convert this into rectangular coordinates \((x, y)\).

Use Conversion Formulas

To convert from polar to rectangular coordinates, use the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). Also, note that \( r^2 = x^2 + y^2 \) and \( \theta = \tan^{-1}(\frac{y}{x}) \).

Substitute \(r = \theta\) in Conversion Formulas

Substitute \( r = \theta \) into the conversion formulas:- For \( x = r \cos(\theta) \), replace \( r \) by \( \theta \): \( x = \theta \cos(\theta) \).- For \( y = r \sin(\theta) \), replace \( r \) by \( \theta \): \( y = \theta \sin(\theta) \). The rectangular form becomes a parametric representation where \( x = \theta \cos(\theta) \) and \( y = \theta \sin(\theta) \).

Sketch the Spiral Graph

In this step, substitute various values of \( \theta \) into the parametric equations to generate specific points on the graph. Trace the path from increasing \( \theta \) values to visualize how the spiral is formed: it winds around the origin, increasing in radius as \( \theta \) increases. This illustrates the Archimedean spiral in polar coordinates as a parametric plot in rectangular coordinates.

Key concepts

Polar Coordinates

Polar coordinates offer a unique way to determine the position of a point on a plane. Unlike the more common rectangular coordinate system, where each point is identified by its horizontal and vertical distance from the origin, polar coordinates use a distance and an angle. Here, each point is defined by two values:

• \(r\): The radial distance from the origin (or pole) to the point.
• \(\theta\): The angle measured from the positive x-axis to the line segment connecting the origin to the point.

It’s like using a compass to describe a point’s location by how far out and what direction from the center point. This makes polar coordinates especially useful in scenarios involving rotation or circular motion.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a familiar system for plotting points on a graph. In this system, each point is described by:

• \(x\): The horizontal position relative to the origin.
• \(y\): The vertical position relative to the origin.

The conversion from polar to rectangular coordinates involves the relationships:

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

This transforms circular and rotational motion into a form where it can be analyzed on a straight grid. By converting from polar to rectangular coordinates, one can appreciate how shapes like spirals map onto the x-y grid.

Archimedean Spiral

The Archimedean spiral is a fascinating curve described by the polar equation \(r = \theta\). Its defining feature is that the distance between successive turns is constant. This spiral begins at the origin (when \(\theta = 0\)) and enlarges outward as \(\theta\) increases.

• In polar terms, it grows proportionally with the angle \(\theta\).
• In rectangular coordinates, this pattern creates an impressively symmetric spiral on a graph.

This characteristic gives it practical applications in various fields, such as engineering and physics, where rotational movement is analyzed.

Parametric Representation

Parametric representation allows complex curves to be expressed through equations that provide a set of conditions for \(x\) and \(y\) independently of each other. For the Archimedean spiral, when converting its polar equation \(r = \theta\) to rectangular form, we use:

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

This approach can break down complex curves into manageable segments. By using various values of \(\theta\), we calculate corresponding \(x\) and \(y\) values, which help sketch the shape accurately over time. This concept is valuable for plotting paths and understanding motion, such as that of objects moving in a spiral trajectory.