Problem 92 - Precalculus

Question

Given \(r=1+\sin (2 \theta)\) and \(r=1-\cos (2 \theta),\) find all points of intersection.

Step-by-Step Solution

Answer

The curves intersect at \(\theta = \frac{3\pi}{8}\) and \(\theta = \frac{7\pi}{8}\).

Step 1: Equate the Equations

Start by setting the two given polar equations equal to each other: \[1 + \sin(2\theta) = 1 - \cos(2\theta)\] Simplify it to obtain:\[\sin(2\theta) = -\cos(2\theta)\]

Step 2: Express Terms in Terms of Tangent

Use the identity that ties sine and cosine to tangent: \(\tan(2\theta) = \frac{\sin(2\theta)}{-\cos(2\theta)} = -1\). This implies:\[\tan(2\theta) = -1\] Thus, \[2\theta = \arctan(-1)\] This occurs whenever \[2\theta = \frac{3\pi}{4} + n\pi\text{, where } n \text{ is an integer}\].

Step 3: Solve for Theta

Solving for \(\theta\), divide all terms by 2:\\[\theta = \frac{3\pi}{8} + \frac{n\pi}{2}\]

Step 4: Verify Intersection Points

Test the solutions for \(\theta\) to see if they produce the same radial distance \(r\) for both equations. Substitute back into either of the original equations to find \(r\):For \(\theta = \frac{3\pi}{8}\), \[r = 1 + \sin(\frac{3\pi}{4}) = 1 + \frac{\sqrt{2}}{2}\]Then check \(\theta = \frac{3\pi}{8} + \frac{\pi}{2}\) to generate additional points.

Step 5: Calculate Additional Points

Using the forms \(\theta = \frac{3\pi}{8} + \frac{\pi}{2}\), calculate:- \(\theta = \frac{7\pi}{8}\), putting it back into, say, \(r = 1 + \sin(2\theta)\) gives:\[r = 1 + \sin(\frac{7\pi}{4}) = 1 + \left(-\frac{\sqrt{2}}{2}\right)\]

Key Concepts

Intersection Points

Intersection points in polar coordinates are fascinating because they involve not just finding where two curves meet, but also discovering the angles and distances from the pole (origin).
Finding intersection points means setting the radial functions equal to each other because both will point to the same location. For our problem, we equate two given polar equations:

• \( r = 1 + \sin(2\theta) \)
• \( r = 1 - \cos(2\theta) \)

By equating them, \( 1 + \sin(2\theta) = 1 - \cos(2\theta) \), we derive \( \sin(2\theta) = -\cos(2\theta) \).
This equation is pivotal as it tells us about their intersection through their angle values \( 2\theta \). By further simplifying, we determine the angles where these curves intersect, eventually leading to finding the exact intersection points on the polar plane.

Trigonometric Identities

Trigonometric identities are essential tools when dealing with polar equations. They help us express relationships between angles and provide us equations to solve angle variables like \( \theta \). In our exercise, recognizing the relationship \( \tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)} \) allows us to effectively find solutions for \( \theta \).
Using the identity that \( \tan(2\theta) = -1 \) implies that

• \( 2\theta = \arctan(-1) \)
• \( 2\theta = \frac{3\pi}{4} + n\pi \) for any integer \( n \)

This formula tells us the possible values for \( 2\theta \), thus informing us of specific angles where the curves intersect.
These identities simplify complex expressions into more manageable terms, making it easier to solve for specific values of \( \theta \). Always remember, understanding these identities expands your ability to solve a broader range of trigonometric problems effectively.

Polar Equations

Polar equations present curves in a plane where each point is determined by a distance from a reference point and an angle from a reference direction. They are different from the Cartesian system and frequently used in scenarios involving rotations or circular movements.
For our problem, we dealt with polar equations like \( r = 1 + \sin(2\theta) \) and \( r = 1 - \cos(2\theta) \), which describe specific types of spirals or loops around the pole. By analyzing these equations, our goal was to find the angles and corresponding distances where they intersect.
Their solution involves understanding that:

• \( r \) gives the distance from the pole
• \( \theta \) gives the angle, crucial for determining position on the polar plane

Substituting values of \( \theta \) back into \( r \) provides precise points of intersection. Mastery of polar equations not only helps in solving these problems but also enhances your capability to visualize and interpret complex curves in polar coordinates.