Problem 2 - 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.\(\left\\{\begin{array}{l}2 r=3 \\ r=1+\cos \theta\end{array}\right.\)

Step-by-Step Solution

Answer

The points of intersection are \( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \).

Step 1 - Express the equations in Cartesian coordinates

Convert the polar equations to Cartesian coordinates. The first equation is given by: \[2r = 3\] Dividing both sides by 2: \[r = \frac{3}{2}\]The second equation is already in standard polar form: \[r = 1 + \cos \theta\]

Step 2 - Sketch the polar graphs

Graph the equations in polar coordinates. For the first equation, \[r = \frac{3}{2}\], this represents a circle centered at the origin with a radius of 1.5 units.For the second equation, \[r = 1 + \cos \theta\], this represents a limaçon with an inner loop.

Step 3 - Set the equations equal to find intersection points

Set the two equations equal to each other to find the points of intersection: \[\frac{3}{2} = 1 + \cos \theta\]Solve for \( \cos \theta \):\[\frac{3}{2} - 1 = \cos \theta\]\[\frac{1}{2} = \cos \theta\]

Step 4 - Find the angle(s) \( \theta \)

Determine the values of \( \theta \) where \( \cos \theta = \frac{1}{2} \).These values are:\[\theta = \frac{\pi}{3}\text{ and }\theta = \frac{5\pi}{3}\]

Step 5 - Calculate the intersection points\( (r, \theta) \)

Use the values of \( \theta \) to find the corresponding \( r \):\[r = \frac{3}{2}\]Thus, the points of intersection \((r, \theta)\) are: \[\left( \frac{3}{2}, \frac{\pi}{3} \right)\text{ and }\left( \frac{3}{2}, \frac{5\pi}{3} \right)\]

Step 6 - Confirm by substitution

Substitute these points back into both original equations to confirm that they satisfy both equations. Both points satisfy the equations.

Key Concepts

polar coordinates

Polar coordinates are an alternative to Cartesian coordinates (x, y). Instead of using horizontal and vertical distances to locate a point, polar coordinates use a radius and an angle. The radius, denoted as r, measures the distance from the origin (center point), while the angle, denoted by \( \theta \), indicates the direction from the positive x-axis.
In polar coordinates, any point is described as \( (r, \theta) \). This system is particularly useful for dealing with curves and equations that have radial symmetry. For instance, a circle centered at the origin with radius \( R \) is simply described as \( r = R \).
Polar coordinates can make complex shapes simpler. Some standard shapes in polar coordinates include circles, spirals, and limaçons.

Cartesian coordinates conversion

Converting between polar and Cartesian coordinates helps in visualizing and solving problems involving different coordinate systems. Here's how you can do it:
From Polar to Cartesian:

• The x-coordinate is given by \( x = r \cos \theta \)
• The y-coordinate is given by \( y = r \sin \theta \)

For the exercise equations, we converted from polar to Cartesian coordinates:
The first equation was \( 2r = 3 \). Dividing by 2, we get \( r = \frac{3}{2} \). In Cartesian coordinates, this would be: \( x^2 + y^2 = \left( \frac{3}{2} \right)^2 \) because \( r^2 = x^2 + y^2 \).
The second equation was already in polar form: \( r = 1 + \cos \theta \). This equation is not as straightforward to convert but graphing it in polar form often provides insight into its behavior.
Converting the equations helps to understand the geometry of the problem without relying solely on plotting graphs.

solving trigonometric equations

Solving trigonometric equations is crucial for finding intersection points in polar coordinates. To find where two polar graphs intersect, set their equations equal and solve for \( \theta \).
In the given exercise, the equations were:

• \( r = \frac{3}{2} \)
• \( r = 1 + \cos \theta \)

Set them equal: \( \frac{3}{2} = 1 + \cos \theta \)
Solve for \( \cos \theta \): \[ \frac{3}{2} - 1 = \cos \theta \] \[ \cos \theta = \frac{1}{2} \]
Find the \( \theta \) values satisfying \( \cos \theta = \frac{1}{2} \): These are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \).
Verify if these angles satisfy both equations. In this case, both angles provided correct intersection points.
Understanding how to manipulate and solve trigonometric equations is vital for intersection problems.

graphing polar equations

Graphing polar equations enables us to visualize points of intersection. For the exercise:
The first equation \( r = \frac{3}{2} \) represents a circle with radius 1.5 units.
The second equation \( r = 1 + \cos \theta \) describes a limaçon with an inner loop.
To graph these polar equations:

• Plot points for various \( \theta \) values and connect them smoothly.
• For \( r = \frac{3}{2} \), every point is 1.5 units away from the origin.
• For \( r = 1 + \cos \theta \), calculate specific \( r \) values for different \( \theta \) values and plot them.

Seeing these graphs clarifies how they overlap and where intersections occur. In this example, the intersections determined by solving trigonometric equations (\( \left( \frac{3}{2}, \frac{\pi}{3} \right) \) and \( \left( \frac{3}{2}, \frac{5\pi}{3} \right) \)) should match up visually on the graph.
Graphing reinforces the solutions and shows the problem's geometry clearly.