Problem 23 - College Trigonometry

Question

In Exercises \(21-30\), find the exact polar coordinates of the points of intersection of graphs of the polar equations. Remember to check for intersection at the pole (origin). $$ r=1-2 \sin (\theta) \text { and } r=2 $$

Step-by-Step Solution

Answer

The polar coordinates of the intersection points are \((2, \frac{7\pi}{6})\) and \((2, \frac{11\pi}{6})\).

Step 1: Set the Equations Equal

We begin by setting the two equations equal to solve for \( \theta \). This will help us find the points of intersection. The equations are \( r = 1 - 2\sin(\theta) \) and \( r = 2 \). Setting them equal gives: \[ 1 - 2\sin(\theta) = 2 \]

Step 2: Solve for \( \sin(\theta) \)

Solve the equation \( 1 - 2\sin(\theta) = 2 \) for \( \sin(\theta) \):\[ 1 - 2\sin(\theta) = 2 \]Subtract 1 from both sides:\[ -2\sin(\theta) = 1 \]Divide both sides by -2:\[ \sin(\theta) = -\frac{1}{2} \]

Step 3: Find \( \theta \) Values

The values of \( \theta \) where \( \sin(\theta) = -\frac{1}{2} \) are:- \( \theta = \frac{7\pi}{6} \)- \( \theta = \frac{11\pi}{6} \)These values are found using the unit circle.

Step 4: Check Intersection at the Pole

To check if the graphs intersect at the pole, set \( r = 0 \) and find \( \theta \) for the first equation:\[ 0 = 1 - 2\sin(\theta) \]Solving gives \( \sin(\theta) = \frac{1}{2} \), which has solutions \( \theta = \frac{\pi}{6}, \frac{5\pi}{6} \). However, these do not satisfy \( r = 2 \) in the second equation.

Step 5: Determine Exact Polar Coordinates

From steps 1 to 4, we determine that the points of intersection occur at \( \theta = \frac{7\pi}{6} \) and \( \theta = \frac{11\pi}{6} \). For both points of intersection, since \( r \) must be 2 in the second equation, the corresponding polar coordinates are:1. \( (2, \frac{7\pi}{6}) \) 2. \( (2, \frac{11\pi}{6}) \)

Key Concepts

Intersection of Graphs

When working with polar graphs, the intersection points represent where the two curves meet. To find the intersection points, we set the equations equal to each other. This allows us to find the common values of \(\theta\), as shown in the original exercise.

• Start by equating the two polar equations, such as \(r=1-2 \ \sin(\theta)\) and \(r=2\).
• Solve for \(\theta\) to find where the intersection occurs.
• Check each obtained solution by substituting back to confirm that the intersection indeed exists on both graphs.

This systematic approach helps in finding exact intersection points, ensuring no solutions are missed. Checking intersects at the origin (or pole) is also vital, but in our case, it did not yield valid points for the second equation.

Polar Equations

Polar equations represent curves based on distance \(r\) from the origin and angle \(\theta\) relative to the positive x-axis. They offer an alternative method to describe shapes and paths, especially appealing in circular or rotational dynamics.

• In contrast to Cartesian coordinates, polar coordinates detail the point's location through angle and radius.
• Equations like \(r = 1 - 2\sin(\theta)\) represent complex curves, often forming loops or rotational shapes.
• The variable \(r\) in polar equations gives the radius, determining the distance from the center or pole.

Polar equations can simplify descriptions of rotational motion or repeating patterns, making them crucial in many fields, from physics to engineering. Understanding these can open unique insights into spatial dynamics and symmetries.

Unit Circle

The unit circle is a fundamental concept in trigonometry. It simplifies finding angles and their trigonometric functions, such as sine and cosine, which are crucial for understanding polar coordinates.

• It's a circle with a radius of 1, centered at the origin of a coordinate plane.
• In the unit circle, the angle \(\theta\) is measured counterclockwise from the positive x-axis.
• The coordinates of a point on this circle are \( (\cos(\theta), \sin(\theta))\), which correspond to the x and y projections.

Utilizing the unit circle helps in quickly finding \(\theta\) for angles where functions \(\sin(\theta)\) and \(\cos(\theta)\) have known values, enabling problem-solving for intersections and analyzing polar graphs efficiently.