Problem 3 - The Calculus with Analytic Geometry

Question

Find the points of intersection of the graphs of the given pair of equations. Draw a sketch of each pair of graphs with the same pole and polar axis.\(\left\\{\begin{array}{l}r=2 \cos \theta \\ r=2 \sin \theta\end{array}\right.\)

Step-by-Step Solution

Answer

The points of intersection are at (\[ \frac{2}{2}\], \[\ \frac{2}{2}\]) with equivalent radius at equal point verifying the intersection in the same pole origin polar axis.

Step 1: Rewrite the Equations in Rectangular Form

First, convert the polar equations to rectangular form. The polar equations are given as:\[ r = 2 \cos \theta \] and \[ r = 2 \sin \theta \].We know that \[ r \cos \theta = x \] and \[ r \sin \theta = y \].Thus, multiplying both sides of each equation by \[r\], we get:\[ x = 2r \cos \theta = 2 \cos \theta \sin \theta\] and \[ y = 2 \sin \theta = 2 \cos \theta \sin \theta\].

Step 2: Set the Rectangular Equations Equal to Each Other

To find the points of intersection, set the two converted equations equal to each other:\[ x = y \> as \sin \theta = \cos \theta \].

Step 3: Solve for Points of Intersection

Given the equation \[ x = y \], substitute back into one of the modified rectangular forms.Since \[ x = y\], we can use the relation from \[ x = y \] as \[ 2 \cos \theta = 2 \cos \theta \].Thus, the points of intersection corresponding to these values are:\[ x = sqrt(2)/2,\ y = sqrt(2)/2\] and usually radius equal to \[r\] intersection points. We found the points of intersection.

Step 4: Verify the Points of Intersection

Finally, verify these points of intersection by substitution back into the original polar equations to ensure both equations hold true.Substituting \[r\] values back into \[ r = 2 \cos \theta \] and \[ r = 2 \sin \theta \]. We get both equations true.

Step 5: Sketch the Graphs

Sketch graphs of \[r=2 \cos \theta\] and \[r=2 \sin \theta\] with the same pole and polar axis. Each graph should intersect at the verified points.

Key Concepts

Polar Equations

Polar equations describe relationships between the variables in polar coordinates, which are represented by \( r \) (the radius or distance from the pole) and \( \theta \) (the angle from the positive x-axis). For example, the polar equation \( r=2 \cos \theta \) represents a circle while \( r=2 \sin \theta \) also represents a circle. These equations are often more intuitive for problems involving angles and rotations, making them useful in trigonometry and calculus. Understanding how to manipulate and visualize these equations is critical for many mathematical problems.

Rectangular Coordinates Conversion

Conversion between polar and rectangular coordinates is often necessary to simplify problems. Rectangular coordinates involve \( x \) and \( y \) coordinates. The conversion formulas are: \( x = r \cos \theta \) and \( y = r \sin \theta \). For instance, for \( r=2 \cos \theta \), we can multiply by \( r \) to get \( r^2 = 2r \cos \theta \), giving us the rectangular form \( x \). Similarly, for \( r=2 \sin \theta \), we convert to \( y \). Understanding these conversions is vital as it often simplifies intersection and graphing tasks.

Intersection Points

Finding intersection points of polar equations involves setting equal the rectangular forms of the equations. Firstly, we convert to rectangular as shown: \( x = 2 \cos \theta \) and \( y = 2 \sin \theta \). Equating these, you get \( x = y \), which simplifies solving for points where the graphs intersect. By substituting back into one of the original equations, we can confirm the coordinates. This fundamental concept helps in solving more complex problems in calculus and physics.

Graph Sketching

Sketching graphs of polar equations helps visualize the intersection points. Begin by plotting the polar equations like \( r=2 \cos \theta \) and \( r=2 \sin \theta \) on the polar grid using their properties. These shapes are circles oriented along the x-axis and y-axis respectively. Ensure to mark intersecting points identified earlier. Visualization is a powerful aid in understanding how these graphs converge and interact.

Trigonometric Identities

Trigonometric identities simplify the manipulation of polar equations. Familiar ones include \( \sin^2 \theta + \ cos^2 \theta = 1 \) and \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). In solving intersections, recognizing that \( \tan \theta = 1 \) leads to \( \theta = \frac{\pi}{4}, \frac{5\pi}{4} \), giving potential intersection angles. Understanding and applying these identities streamlines equation solving and graph sketching.