Problem 69 - Algebra and Trigonometry

Question

Graph each pair of polar equations on the same polar grid. Find the polar coordinates of the point(s) of intersection and label the point(s) on the graph. $$ r=\sin \theta ; r=1+\cos \theta $$

Step-by-Step Solution

Answer

The points of intersection are (\( \frac{\sqrt2}{2} , \ \frac{\pi}{4} \)) and (\( \frac{\sqrt2}{2} , \ \frac{7\pi}{4} \)).

Step 1: Understand the Polar Equations

The given polar equations are: 1. \(r = \sin \theta\) 2. \(r = 1 + \cos \theta\)

Step 2: Graph the Equations

Plot each equation on the same polar grid. The equation \(r = \sin \theta\) represents a circle passing through the origin, centered at \((0, \frac{1}{2})\). The equation \(r = 1 + \cos \theta\) represents a cardioid.

Step 3: Set the Equations Equal to Find Intersection

To find the points of intersection, set \( \sin \theta = 1 + \cos \theta \).

Step 4: Isolate \( \theta \)

Rearrange the equation to \( \sin \theta - \cos \theta = 1 \). Use trigonometric identities to simplify further. \( \sin \theta = 1 + \cos \theta \) Using identities: \( \sin \theta - \cos \theta = 1 \) Divide by \sqrt2: \( \frac{\sin \theta - \cos \theta}{\sqrt2} = \frac{1}{\sqrt2} \) That simplifies to \( \sin(\theta - \frac{\pi}{4})= \frac{1}{\sqrt2} \). So, \( \theta = \frac{\pi}{4} + 2n\pi\) or \( \theta = \frac{7\pi}{4} + 2n\pi\).

Step 5: Find r for Intersection Points

For \( \theta = \frac{\pi}{4}\): \( r = \sin \frac{\pi}{4} = \frac{\sqrt2}{2} \). For \( \theta = \frac{7\pi}{4} \): \( r = \sin \frac{7\pi}{4} = \frac{-\sqrt2}{2} \). So, \( r = \sqrt2 \frac{1+\frac{\sqrt2}{2}}{1+\frac{\sqrt2}{2}} \). Thus, points are: (\( \frac{\sqrt2}{2} , \ \frac{\pi}{4} \) and \( \frac{\sqrt2}{2} , \ \frac{7\pi}{4} \)

Step 6: Label Points on the Graph

Plot and label the intersection points found in Step 5: ( \( \frac{\sqrt2}{2}, \ \frac{\pi}{4} \) and \( \frac{\sqrt2}{2}, \ \frac{7\pi}{4} \)). These points should be marked clearly on the polar grid.

Key Concepts

Polar Coordinates

Polar coordinates offer a different way to describe the position of points compared to the Cartesian coordinate system. Instead of using \( (x, y) \) pairs, polar coordinates \( (r, \theta) \) consist of:

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

This system is particularly useful for dealing with circular and spiral shapes, where distances and angles from a central point simplify the equations. In essence: the radius \( r \) can be positive or negative, where negative \( r \) implies a direction opposite to \( \theta \). The angle \( \theta \) can range from 0 to 2\( \pi \) for a full circle. Understanding polar coordinates is fundamental for graphing and analyzing polar equations effectively.

Graphing Polar Equations

To graph polar equations, it’s essential to understand the shape each equation represents. Let’s consider the equations provided:
\(r = \sin \theta \): This is a circle that passes through the origin and is centered at \( (0, \frac{1}{2}) \). As \( \theta \) increases from 0 to 2\( \pi \), the value of \( r \) changes accordingly, sketching out the circle.
\(r = 1 + \cos \theta \): This represents a cardioid, a heart-shaped curve that touches the origin. It occurs because, at \( \theta = \pi \), \( r = 0 \).
To graph these, plot several points for varying \( \theta \) values and connect them smoothly. Using polar grid paper can help visualize and plot these shapes accurately. Practice plotting these will bolster confidence in working with different polar equations.

Intersection Points

Finding intersection points in polar equations requires setting the equations equal to each other and solving for \( \theta \). For this exercise:

• Equate \( \sin \theta \) to \(1 + \cos \theta \).
• Simplify to find the common angles: \( \sin \theta - \cos \theta = 1 \).
• Use trigonometric identities to solve further. You get: \( \sin( \theta - \frac{ \pi }{4} ) = \frac {1}{ \sqrt{2} } \).
• Solve this to find \( \theta = \frac{ \pi }{ 4 } \) and \( \frac {7 \pi } {4 } \).

Then, substitute back into either original equation to find \( r \). This results in points:

• \( \frac{ \sqrt{ 2 } }{2}, \frac { \pi } {4 } \)
• \( \frac{ormalsize \sqrt{ 2 } } {2}, \frac{ 7 \pi } { 4 } \)

These are the intersection points to be clearly marked on the polar grid, giving a visual and mathematical understanding of where the graphs meet.