Problem 11 - Calculus: Early Transcendentals

Question

Plot the polar equations \(r=\sin \theta\) and \(r=\cos \theta\) and comment on their similarities. (If you get stuck on how to plot these, you can multiply both sides of each equation by \(r\) and convert back to rectangular coordinates).

Step-by-Step Solution

Answer

Both graphs are circles with radius \(\frac{1}{2}\), symmetric about the line \(y = x\).

Step 1: Convert to Rectangular Coordinates

To convert the polar equations to rectangular coordinates, use the identities \[ r = \sqrt{x^2 + y^2}, \ x = r \cos \theta, \ y = r \sin \theta \].For \( r = \sin \theta \), multiply both sides by \( r \):\[ r^2 = r \sin \theta \]Substitute \( r^2 = x^2 + y^2 \) and \( r \sin \theta = y \):\[ x^2 + y^2 = y \]Rearrange to get:\[ x^2 + (y - \frac{1}{2})^2 = \frac{1}{4} \]. This is a circle centered at \((0, \frac{1}{2})\) with radius \(\frac{1}{2}\).

Step 2: Convert Second Equation

For \( r = \cos \theta \), multiply both sides by \( r \):\[ r^2 = r \cos \theta \]Substitute \( r^2 = x^2 + y^2 \) and \( r \cos \theta = x \):\[ x^2 + y^2 = x \]Rearrange to get:\[ (x - \frac{1}{2})^2 + y^2 = \frac{1}{4} \]. This represents a circle centered at \((\frac{1}{2}, 0)\) with radius \(\frac{1}{2}\).

Step 3: Analyze the Graphs

Now that we have the equations of the two circles, draw them in the Cartesian coordinate system. The first circle is centered at \((0, \frac{1}{2})\) and the second circle at \((\frac{1}{2}, 0)\). Both circles have the same radius \(\frac{1}{2}\).

Step 4: Comment on Similarities

The two graphs are similar because they are both circles with the same radius, and their centers are symmetric about the line \( y = x \), making them symmetrical to each other in the polar coordinate system.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to specify a point in a plane using an ordered pair of perpendicular lines. The two axes, named x and y, intersect at a point called the origin. Each point in the plane is described by a pair of values,

• x-coordinate: Indicates the horizontal position.
• y-coordinate: Indicates the vertical position.

Rectangular coordinates are particularly useful for graphing and analyzing algebraic equations, as they allow easy visualization on a two-dimensional plane. In the exercise, converting polar equations to rectangular form involves using identities that relate these two concepts. This transformation is important because it allows the depiction of polar curves using familiar x-y notation, aiding in understanding and graphing complex equations with ease.
By understanding the relationship between polar and rectangular equations, students can apply this knowledge to accurately graph and analyze different graph shapes.

Graphing Polar Equations

Polar coordinates differ from rectangular coordinates in that they are based on a point's distance from the origin and its angle from the positive x-axis. In polar coordinates,

• r: Represents the radius, or distance from the origin.
• \(\theta\): Is the angle in radians from the positive x-axis.

When graphing polar equations like \(r = \sin\theta\) and \(r = \cos\theta\), you start by determining values of \(\theta\) and calculating the corresponding r. These values are then plotted on a polar graph.
For example, in the exercise, converting these equations to Cartesian by multiplying both sides by r makes it easier to see that they represent circles. The advantage of using polar coordinates is that they sometimes simplify the appearance of certain curves, particularly those involving symmetry about the origin or a particular line.
Understanding how to convert and plot polar equations gives insight into unique geometric shapes and patterns, offering a broader perspective on mathematical functions beyond traditional rectangular plotting.

Symmetry in Graphs

Symmetry plays a crucial role in understanding graphs, as it provides information about the overall shape and behavior of graphs. In the context of the given exercise and its solutions, recognizing symmetry helps simplify graphing and analysis. The concept of symmetry can manifest in several forms:

• Symmetry about the x-axis: If the graph on one side of the x-axis mirrors the other.
• Symmetry about the y-axis: If the graph to the left of the y-axis mirrors the right.
• Symmetry about the origin: If the graph is unchanged when rotated 180 degrees about the origin.

In the polar equations given, the graphs of \(r = \sin\theta\) and \(r = \cos\theta\) exhibit symmetry concerning the line \(y = x\). Both result in circles with the same radius, demonstrating symmetry about the centers of these circles. Recognizing these forms of symmetry allows for quicker graphing and validation of the transformed equations, offering visual and mathematical insight into how the curves relate to each other and simplifying the analysis considerably.