Problem 93 - Precalculus

Question

Given \(r=2+\sin (4 \theta)\) and \(r=1,\) find the angles of all points of intersection.

Step-by-Step Solution

Answer

The angles are \( \theta = \frac{3\pi}{8} \) and \( \frac{7\pi}{8} \).

Step 1: Equating the Equations

Since we are given two equations in polar form, we need to find the angles \( \theta \) where both are equal. Thus, we set the equations equal: \[ 2 + \sin(4\theta) = 1. \] This simplifies to \( \sin(4\theta) = -1 \).

Step 2: Finding the Angles

The sine function equals \(-1\) at specific angles. For sine, this occurs at \[ 4\theta = \frac{3\pi}{2} + 2k\pi \] where \( k \) is an integer. Solving for \( \theta \), we get \[ \theta = \frac{3\pi}{8} + \frac{k\pi}{2}. \] We must determine the values of \( \theta \) within the typical polar coordinate range \( [0, \pi) \).

Step 3: Determining Specific Values of \( \theta \)

Substitute integer values of \( k \):1. For \( k = 0, \theta = \frac{3\pi}{8} \).2. For \( k = 1, \theta = \frac{3\pi}{8} + \frac{\pi}{2} = \frac{7\pi}{8} \).3. For \( k = 2, \theta = \frac{3\pi}{8} + \pi = \frac{11\pi}{8} \) (out of range).Thus, we have valid \( \theta \) values at \( \frac{3\pi}{8} \) and \( \frac{7\pi}{8} \).

Key Concepts

Angle of Intersection

When dealing with polar coordinates, the angle of intersection is a crucial concept to understand. It refers to the angles where two curves, expressed in polar equations, intersect. In the given exercise, we have two polar equations. The task is to find the angles \( \theta \) where these curves meet. We do this by equating the two equations and solving for \( \theta \).
This process often involves determining the conditions under which both equations provide a common point. By equating the expressions and solving for \( \theta \), we identify these intersections, which are vital for understanding the spatial relation between the curves. In polar coordinates, the angle often determines not just where, but how curves intersect and interact.

• Relevant to studying curve behaviors in polar coordinates
• Helps in visualizing how different curves meet at points
  

Understanding the angle of intersection gives insight into the structure and layout of curves on a polar graph, enriching your ability to analyze complex patterns and geometries.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, or tangent. Solving these equations is essential in many areas of mathematics, including when working with polar coordinates. In the context of the given exercise, solving the trigonometric equation \( \sin(4\theta) = -1 \) was a key step.
This equation ultimately determines the specific angles \( \theta \) where the graphs intersect. By setting the sine of an angle equal to a known value (like -1 in this case), you can solve for the angle itself by considering the unit circle or known identity values.

• Often involve functions like sine and cosine
• Solutions often rely on understanding key angle values from the unit circle
• Important for finding specific solutions within defined ranges
  

Trigonometric equations are powerful tools in solving and understanding the dynamics of polar curves, making them an essential skill for students to master.

Sine Function

The sine function is one of the fundamental trigonometric functions, often used to model periodic phenomena. In polar coordinates, the sine function plays a crucial role in defining and solving for intersections of curves. In our exercise, once we reach the equation \( \sin(4\theta) = -1 \), understanding the sine function helps us find the specific angles involved.
The sine function value \(-1\) indicates that we are looking at an angle where the sine reaches its minimum on the unit circle. This typically occurs at angles which are odd multiples of \( \frac{3\pi}{2} \), guiding us towards solutions for \( \theta \).

• Key values: \( \sin(0) = 0 \), \( \sin(\frac{\pi}{2}) = 1 \), \( \sin(\pi) = 0 \), \( \sin(\frac{3\pi}{2}) = -1 \)
• Essential for solving problems involving wave-like patterns
• Integral to understanding curves in polar form
  

By mastering the sine function, you will enhance your ability to solve trigonometric equations, analyze periodic trends, and understand polar graphs more deeply.