Problem 95 - Precalculus

Question

In calculus, when we need to find the area enclosed by two polar curves, the first step consists of finding the points where the curves coincide. Find the points of intersection of the given curves. $$r=4 \sin \theta \quad \text { and } \quad r=4 \cos \theta$$

Step-by-Step Solution

Answer

The curves intersect at \((r, \theta) = \left( 2\sqrt{2}, \frac{\pi}{4} \right)\).

Step 1: Set Equations Equal to Each Other

To find the points of intersection, equate the two given polar equations: \[ 4 \sin \theta = 4 \cos \theta \] Divide both sides by 4 to simplify:\[ \sin \theta = \cos \theta \]

Step 2: Solve the Trigonometric Equation

To solve \(\sin \theta = \cos \theta\), divide both sides by \(\cos \theta\) (assuming \(\cos \theta eq 0\)):\[ \tan \theta = 1 \]The solution to \(\tan \theta = 1\) is \(\theta = \frac{\pi}{4} + n\pi\), where \(n\) is any integer.

Step 3: Find Radius at Intersection Points

Substitute \(\theta = \frac{\pi}{4}\) into one of the original equations to find \(r\):\[ r = 4 \sin \left( \frac{\pi}{4} \right) = 4 \left( \frac{\sqrt{2}}{2} \right) = 2\sqrt{2} \]Thus, one point of intersection is \((r, \theta) = \left( 2\sqrt{2}, \frac{\pi}{4} \right)\).

Step 4: Consider Other Solutions for θ

Since \(\theta = \frac{\pi}{4} + n\pi\), check for the next possible angle which keeps \(r\) positive:\[ \theta = \frac{5\pi}{4} \]Substitute this \(\theta\) back into one of the original equations to find \(r\):\[ r = 4 \sin \left( \frac{5\pi}{4} \right) = 4 \left( -\frac{\sqrt{2}}{2} \right) = -2\sqrt{2} \]Negative \(r\) indicates the same point as \(\left( 2\sqrt{2}, \frac{\pi}{4} \right)\), in a different direction, which doesn't give a new intersection point.

Key Concepts

Intersection of Polar Curves

To find the points where two polar curves intersect, we need to set their equations equal to each other. Polar coordinates are a way of representing points in the plane, where each point is defined by a radius and an angle. In this exercise, we have two polar curves:

• \(r = 4 \sin \theta\)
• \(r = 4 \cos \theta\)

By setting these two equations equal, \(4 \sin \theta = 4 \cos \theta\), and simplifying, \(\sin \theta = \cos \theta\), we aim to find the values of \(\theta\) where both curves share the same radius \(r\). Solving \(\tan \theta = 1\) tells us that the intersection occurs at specific angles: \(\theta = \frac{\pi}{4} + n\pi\), which repeats for every integer \(n\). However, we must ensure that the found angles yield positive radius values for a legitimate point of intersection.

Trigonometric Identities

Trigonometric identities play a pivotal role in solving equations involving polar curves. In our exercise, we encountered the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Using such identities simplifies solving for \(\theta\) so we can determine the intersection of curves.Firstly, identifying that \(\tan \theta = 1\), suggests that \(\theta\) occurs at angles where sine and cosine are equal, like \(\theta = \frac{\pi}{4}\). Recognizing these expressions helps to deduce multiple solutions for \(\theta\), as trigonometric functions are periodic. Hence, the addition of \(n\pi\) arises because tangent has a period of \(\pi\). This understanding underscores the significance of well-known trigonometric angles and helps in finding repeated occurrences of similar solutions.Finally, when you understand how to maneuver with these identities, fixing intersections or even integrating areas becomes much smoother, reinforcing the necessity of memorizing core identities.

Area Between Polar Curves

Finding the area between two polar curves involves understanding both their intersections and the nature of polar integration. Once the intersection points are known, the next step would typically be to integrate the area enclosed between the curves.
The formula to find the area between two polar curves \(r_1(\theta)\) and \(r_2(\theta)\) from \(\theta = a\) to \(\theta = b\) is provided as:\[\text{Area} = \frac{1}{2} \int_a^b \left( r_1(\theta)^2 - r_2(\theta)^2 \right) d\theta\]
This formula ensures you're subtracting the inner curve's squared radius from the outer curve's squared radius. Essentially, it measures the segment between the two curves.
To accurately compute the area, it's crucial to diagnose and correctly account for intersections since these dictate the bounds of integration. Misjudging this step could lead to incomplete or incorrect area computation. Through understanding and correctly applying this method, students can proficiently handle tasks involving areas between polar curves, significantly aided by the clear comprehension of intersection and trigonometric principles.