Problem 34 - Thomas Calculus

Question

Find the points of intersection of the pairs of curves in Exercises \(31-38\) . $$ r=\cos \theta, \quad r=1-\cos \theta $$

Step-by-Step Solution

Answer

Intersection points: \( \left( \frac{1}{4}, \frac{\sqrt{3}}{4} \right) \) and \( \left( \frac{1}{4}, -\frac{\sqrt{3}}{4} \right) \).

Step 1: Convert to Cartesian Coordinates

To find the points of intersection of the polar equations, convert them to Cartesian coordinates. The equations are given as: \[ r = \cos \theta \quad \text{and} \quad r = 1 - \cos \theta \] To convert, use the formulas: \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] First substitute \( r = \cos \theta \): \[ x = (\cos \theta) \cos \theta = \cos^2 \theta \] \[ y = (\cos \theta) \sin \theta = \cos \theta \sin \theta \]For \( r = 1 - \cos \theta \): \[ x = (1-\cos \theta) \cos \theta = \cos \theta - \cos^2 \theta \] \[ y = (1-\cos \theta) \sin \theta = \sin \theta - \cos \theta \sin \theta \].

Step 2: Set Polar Equations Equal

Set the polar equations equal to find \( \theta \) for intersecting points: \[ r = \cos \theta = 1 - \cos \theta \] Solving for \( \cos \theta \): \[ \cos \theta + \cos \theta = 1 \] \[ 2\cos \theta = 1 \] \[ \cos \theta = \frac{1}{2} \]

Step 3: Find Theta Values

Solve for \( \theta \) when \( \cos \theta = \frac{1}{2} \). From trigonometric identities, we know \( \cos \theta = \frac{1}{2} \) at \(\theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) within the range \(0 \leq \theta < 2\pi \).

Step 4: Calculate Intersection Points

Substitute \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{5\pi}{3} \) into the original polar equations to find the radial distance \( r \). 1. For \( \theta = \frac{\pi}{3} \): - \( r = \cos(\frac{\pi}{3}) = \frac{1}{2} \) - Intersection point: \( \left( \frac{1}{2}, \frac{\pi}{3} \right) \)2. For \( \theta = \frac{5\pi}{3} \): - \( r = \cos(\frac{5\pi}{3}) = \frac{1}{2} \) - Intersection point: \( \left( \frac{1}{2}, \frac{5\pi}{3} \right) \)

Step 5: Express in Cartesian Coordinates

Convert polar intersection points to Cartesian coordinates for verification:1. For \( \left( \frac{1}{2}, \frac{\pi}{3} \right) \): - \( x = \frac{1}{2} \cdot \cos \left( \frac{\pi}{3} \right) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \) - \( y = \frac{1}{2} \cdot \sin \left( \frac{\pi}{3} \right) = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \) - Intersection point: \( \left( \frac{1}{4}, \frac{\sqrt{3}}{4} \right) \)2. For \( \left( \frac{1}{2}, \frac{5\pi}{3} \right) \): - \( x = \frac{1}{2} \cdot \cos \left( \frac{5\pi}{3} \right) = \frac{1}{4} \) - \( y = \frac{1}{2} \cdot \sin \left( \frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{4} \) - Intersection point: \( \left( \frac{1}{4}, -\frac{\sqrt{3}}{4} \right) \)

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane through a simple system involving a radius and an angle. Rather than using an x and y axis, polar coordinates use a single origin point and measure points based on a distance (r) and an angle (θ) from that origin.

• Radial Distance (r): This is the distance from the origin to the point.
• Angle (θ): This angle, measured in radians, is taken from the positive x-axis and moves counterclockwise.

Polar coordinates are particularly useful for solving mathematical problems involving circles and spirals where Cartesian coordinates (which use perpendicular x and y axes) might be cumbersome. In the context of curves such as the ones given, expressing them in terms of polar coordinates often simplifies the understanding and solving process.

Cartesian Coordinates

Cartesian coordinates are possibly the most familiar coordinate system, utilizing horizontal (x) and vertical (y) axes to represent a point in a two-dimensional space. This system is highly intuitive for representing straight lines and is very helpful in algebra and geometry.

• Conversion from Polar to Cartesian: To change from polar \(r, \theta\) to Cartesian \((x, y)\), the formulas used are \(\begin{align*}x &= r \cos \theta \y &= r \sin \theta\end{align*}\)
• Implementation: Once a polar equation is converted to Cartesian form, it becomes easier to plot and understand some mathematical problems visually.

In the original exercise, converting polar equations to Cartesian helps in verifying the accuracy of the intersections found in the polar system. Even though both systems are different, they eventually communicate the same point in space but from different perspectives.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions like sine, cosine, and tangent, which hold true for all angles. These identities help simplify complex trigonometric equations and are pivotal in solving them across different problems.

• Basic Identities: Some commonly used are \(\sin^2 \theta + \cos^2 \theta = 1\), which always holds true and \(\cos(2\theta)=\cos^2\theta-\sin^2\theta\).
• Using Identities in Intersections: In the exercise, knowing that \(\cos \theta = \frac{1}{2}\) leads to two potential angles, \(\theta = \frac{\pi}{3}\) and \(\theta = \frac{5\pi}{3}\), is derived using these identities.

Understanding trigonometric identities allows you to see patterns and relations that are not immediately obvious, significantly aiding in problem-solving. In the context of the original exercise, these identities help find the specific angle values (θ) where the two curves intersect, allowing for a complete solution of the problem.