Question

Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. \(r=1-\sin \theta\) and \(r=1+\cos \theta\)

Short answer

Question: Determine the intersection points of the polar coordinate graphs \(r = 1 - \sin\theta\) and \(r = 1 + \cos\theta\). Answer: The intersection points are \((0.2929,3\pi/4)\) and \((1.7071,7\pi/4)\).

Step-by-step solution

First, find intersection points by equating both the equations

To find the intersection points of the polar coordinate graphs, \(r_1=1-\sin\theta\) and \(r_2=1+\cos\theta\), we will solve the equation \(1-\sin\theta=1+\cos\theta\) $$1-\sin\theta=1+\cos\theta$$ $$\cos\theta+\sin\theta=0$$ Now, we will solve for \(\theta\): $$\sin\theta=-\cos\theta$$ $$\cot\theta=-1$$ From this equation, we get two possible values of \(\theta\) $$\theta=3\pi/4 , 7\pi/4$$ Now we compute the corresponding r-values: $$r_1=1-\sin(3\pi/4)=1-\frac{\sqrt{2}}{2}\approx 0.2929$$ $$r_1=1-\sin(7\pi/4)=1+\frac{\sqrt{2}}{2}\approx 1.7071$$ So, we found two intersection points: \((0.2929,3\pi/4)\) and \((1.7071,7\pi/4)\). But there may be more intersection points as polar coordinates can represent a point with different angle values. To identify all the intersection points, we need to use a graphing utility.

Use a graphing utility to find all intersection points

By using a graphing tool, we can see if there are any other intersection points corresponding to different angle values. After checking the graphs and analyzing them, it turns out that the two points we obtained are the only intersection points of the polar coordinate graphs \(r = 1 - \sin\theta\) and \(r = 1 + \cos\theta\). Therefore, the intersection points of the given polar coordinate graphs are \((0.2929,3\pi/4)\) and \((1.7071,7\pi/4)\).

Key concepts

Intersection Points

In polar coordinates, finding intersection points between curves involves determining where the different curve equations result in the same point. Consider two polar equations like \(r_1 = 1 - \sin \theta\) and \(r_2 = 1 + \cos \theta\). An intersection point is where the radius from the origin, \(r\), and the angle, \(\theta\), are the same for both equations.

To find these points analytically, equate the equations: \( 1 - \sin \theta = 1 + \cos \theta \). Solve the resulting equation \( \cos \theta + \sin \theta = 0 \), which translates to \( \cot \theta = -1 \). This equation simplifies to yield multiple angle solutions, such as \(\theta = 3\pi/4\) and \(\theta = 7\pi/4\).

Remember, in polar coordinates, a point can have multiple \(\theta\) values. This property makes it helpful to use visual aids or graphing utilities to confirm all intersection points.

Graphing Utility

A graphing utility is a helpful tool that offers a visual representation of mathematical concepts. It provides a more intuitive understanding of problems in polar coordinates and is especially useful when identifying intersection points between curves.

By graphing the polar equations \(r_1 = 1 - \sin \theta\) and \(r_2 = 1 + \cos \theta\), you can visually check the intersection points determined analytically. While you might have found points such as \((0.2929,3\pi/4)\) and \((1.7071,7\pi/4)\), the graphing utility helps confirm that no other intersection points exist. This visual method ensures all possible solutions are considered, taking advantage of the periodic nature of trigonometric functions in polar graphs.

With the graphing utility, spotting overlapped curves becomes apparent, ensuring comprehensive analysis beyond algebraic solutions.

Trigonometric Equations

Trigonometric equations play a crucial role in solving problems related to polar coordinates. These equations involve trigonometric functions such as sine and cosine, which are inherently periodic.

In the exercise given, solving \( \cos \theta + \sin \theta = 0 \) is a critical step. To solve this, you substitute \( \sin \theta \) as \(- \cos \theta\), making \( \cot \theta = -1 \). From here, you determine values for \(\theta\) that satisfy this condition, which are \(\theta = 3\pi/4\) and \(\theta = 7\pi/4\).

Understanding how to manipulate trigonometric equations lets you draw connections between different angles that yield the same polar coordinate point. This feature is vital since it reflects the symmetry and periodic characteristics of trigonometric functions in polar equations.

• Records solutions through manipulation of trigonometric identities.
• Pays attention to periodicity to capture all relevant angle values.