Chapter 10: Problem 67
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. $$ r=8 \cos \theta ; r=2 \sec \theta $$
Short Answer
Expert verified
The points of intersection are (4, \( \frac{\pi}{3} \)), (-4, \( \frac{2 \pi}{3} \)), (4, \( \frac{5 \pi}{3} \)), and (-4, \( \frac{4 \pi}{3} \)).
Step by step solution
01
- Understand the Polar Equations
The given polar equations are \( r = 8 \, \cos \, \theta \) and \( r = 2 \, \sec \, \theta \). Recall that \( \sec \, \theta = \frac{1}{\cos \, \theta} \).
02
- Convert the Second Equation
Rewrite the second equation in terms of cosine. \( r = 2 \, \sec \, \theta \) becomes \( r = 2 / \cos \, \theta \).
03
- Find Common Polar Coordinates
Set \( r \) from both equations equal to find the intersection points: \[ 8 \, \cos \, \theta = \frac{2}{\cos \, \theta} \].
04
- Solve for \( \cos \, \theta \)
Multiply both sides by \( \cos \, \theta \): \[ 8 \, \cos^2 \, \theta = 2 \]. So, \[ \cos^2 \, \theta = \frac{2}{8} = \frac{1}{4} \], \( \cos \, \theta = \pm \frac{1}{2} \).
05
- Determine Corresponding Angles
\( \cos \, \theta = \frac{1}{2} \) occurs at \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5 \pi}{3} \). \( \cos \, \theta = -\frac{1}{2} \) occurs at \( \theta = \frac{2 \pi}{3} \) and \( \theta = \frac{4 \pi}{3} \).
06
- Compute \( r \) for Intersection Points
Using the initial equation \( r = 8 \, \cos \, \theta \), calculate \( r \) for each angle.When \( \theta = \frac{\pi}{3} \) or \( \theta = \frac{5 \pi}{3} \): \[ r = 8 \, \cos \frac{\pi}{3} = 8 \times \frac{1}{2} = 4 \].When \( \theta = \frac{2 \pi}{3} \) or \( \theta = \frac{4 \pi}{3} \): \[ r = 8 \, \cos \frac{2 \pi}{3} = 8 \times -\frac{1}{2} = -4 \].
07
- Identify Intersection Points
The points of intersection are: \( (4, \frac{\pi}{3}) \), \( (-4, \frac{2 \pi}{3}) \), \( (4, \frac{5 \pi}{3}) \), and \( (-4, \frac{4 \pi}{3}) \).
08
- Graph the Polar Equations
Graph both equations on the same polar grid and label the intersection points identified in the previous step.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
polar equations
Polar equations describe a relationship between the distance from the origin (called the radial coordinate, denoted by \( r \)) and the angle \( \theta \) measured from the positive x-axis. Unlike Cartesian coordinates which use (x, y) pairs, polar coordinates use (r, \( \theta \)) pairs. This provides a convenient way to describe curves and shapes that have symmetry around a central point. For example, the equation \( r = 8 \cos \theta \) describes a circle centered at (4, 0) on the polar grid.
polar grid
A polar grid consists of concentric circles and lines emanating from the origin, dividing the plane into different angles. The concentric circles represent the changing values of \( r \), and the lines represent different angles, \( \theta \). This grid is essential for graphing polar equations, as it allows you to convert between polar and Cartesian coordinates easily. On a polar grid, the point with coordinates \( (r, \theta) \) is plotted by moving \( r \) units away from the origin at an angle \( \theta \).
intersection points
Finding intersection points between polar equations involves setting the equations equal to each other and solving for \( \theta \). Once the values of \( \theta \) are known, substitute them back into one of the equations to find the corresponding \( r \) values. These (\( r, \theta \)) pairs are the intersection points. For example, given the equations \( r = 8 \cos \theta \) and \( r = 2 \sec \theta \), we find the common angles \( \theta \) by solving for \( \cos \theta \). This yields intersection points like \( (4, \frac{\pi}{3}) \).
cosine function
The cosine function, denoted as \( \cos \theta \), relates the angle \( \theta \) to the adjacent side and hypotenuse of a right triangle in trigonometry. In polar equations, cosine helps describe the radial distance \( r \) based on the angle \( \theta \). For example, in the equation \( r = 8 \cos \theta \), \( r \) depends on the cosine of the angle. Specific values of \( \cos \theta \), such as \( \cos \theta = \pm \frac{1}{2} \), help determine important angles where intersection points may exist. The cosine function is periodic with a period of \( 2\pi \), which means it repeats its values every \( 360^\circ \) or \( 2\pi \) radians.