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Chapter 10: Problem 38

Find all points of intersection of the given curves. $$ r=1+\cos \theta, \quad r=1-\sin \theta $$

Short Answer

Expert verified
The points of intersection are \( \left(1 - \frac{\sqrt{2}}{2}, \frac{3\pi}{4}\right) \) and \( \left(1 + \frac{\sqrt{2}}{2}, \frac{7\pi}{4}\right) \) in polar coordinates.

Step by step solution

01

Equate the Equations

We have two polar equations: \( r = 1 + \cos \theta \) and \( r = 1 - \sin \theta \). To find the points of intersection, first equate these equations: \[ 1 + \cos \theta = 1 - \sin \theta \].
02

Simplify the Equation

Simplify the equation by subtracting 1 from both sides: \( \cos \theta = - \sin \theta \).
03

Solve for \( \theta \)

To find the angle \( \theta \), divide both sides of the equation by \( \cos \theta \) (assuming \( \cos \theta eq 0 \) to avoid division by zero): \( 1 = -\tan \theta \) This implies \( \tan \theta = -1 \).
04

Determine the Angles

The equation \( \tan \theta = -1 \) holds true for angles \( \theta = \frac{3\pi}{4} \) and \( \theta = \frac{7\pi}{4} \) within the interval \([0, 2\pi)\).
05

Find Corresponding \( r \) Values

For each value of \( \theta \), substitute back into one of the original equations to find \( r \). Let's use \( r = 1 + \cos \theta \):- For \( \theta = \frac{3\pi}{4} \), \( \cos \left( \frac{3\pi}{4} \right) = -\frac{1}{\sqrt{2}} \) or equivalently, \( -\frac{\sqrt{2}}{2} \), thus \( r = 1 - \frac{\sqrt{2}}{2} \).- For \( \theta = \frac{7\pi}{4} \), \( \cos \left( \frac{7\pi}{4} \right) = \frac{1}{\sqrt{2}} \) or equivalently, \( \frac{\sqrt{2}}{2} \), thus \( r = 1 + \frac{\sqrt{2}}{2} \).
06

List the Points of Intersection

The points of intersection in polar coordinates \((r, \theta)\) are:- \( \left(1 - \frac{\sqrt{2}}{2}, \frac{3\pi}{4}\right) \)- \( \left(1 + \frac{\sqrt{2}}{2}, \frac{7\pi}{4}\right) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intersection of Curves
To find where two curves intersect, we look for points that are shared between the curves. In polar coordinates, curves are defined using polar equations where each point is determined by a distance from the origin, r, and an angle, \(\theta\), from the polar axis.
For the curves described by the equations \(r = 1 + \cos \theta\) and \(r = 1 - \sin \theta\), the intersection points are the values of \(\theta\) where both curves yield the same value for \(r\).
This involves equating the parallel forms of \(r\) from both equations and solving for \(\theta\).
  • Equating the equations gives the condition for intersection.
  • Solving these equations yields the angles \(\theta\) where intersections occur.
  • Substituting these \(\theta\) values back into the polar equations gives the radial distance, \(r\), for each intersection point.

This provides the complete set of intersection points in the polar coordinate system.
Trigonometric Functions
Trigonometric functions describe relationships between angles and lengths in triangles. The most common functions are sine, cosine, and tangent, which relate an angle to the ratios of sides of a right triangle.
In polar coordinates, these functions are used to convert between different forms of equations as they help express points in relation to the angle \(\theta\).
In this exercise, the curve equations use the functions \(\cos \theta\) and \(\sin \theta\):
  • These functions oscillate between -1 and 1, describing the circular nature of the polar curves.
  • The equation \(1 + \cos \theta\) moves the baseline of cosine's wave up by 1, affecting the curve's expansion.
  • Similarly, \(1 - \sin \theta\) shifts the sine wave downward, influencing the curve's layout.

These manipulations of sine and cosine are crucial in determining how the polar curves behave and intersect.
Polar Equations
A polar equation expresses the relationship between the radial coordinate, \(r\), and the angular coordinate, \(\theta\). Unlike Cartesian equations, which use \(x\) and \(y\), polar equations naturally accommodate curves that are circular or spiral in nature.
In the example problem, polar equations give us a deeper insight into shapes often resembling circles, spirals, and flower-like structures.
To solve problems involving polar equations like \(r = 1 + \cos \theta\) and \(r = 1 - \sin \theta\), we:
  • Analyze the structure of the equation to understand the curve it defines.
  • Identify general properties of polar graphs, such as symmetry and periodicity, which simplify complex calculations.
  • Use trigonometric identities and algebra to find intersections and other characteristics of the equations.

The application of polar equations is wide-ranging, including modeling physical phenomena and computer-generated graphics.

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