Problem 14 - The Calculus with Analytic Geometry

Question

Find a measurement of the angle between the tangent lines of the given pair of curves at all points of intersection.\(\left\\{\begin{array}{l}r=3 \cos \theta \\\ r=1+\cos \theta\end{array}\right.\)

Step-by-Step Solution

Answer

Find the intersection points, derive the tangent slopes, and calculate the angle using the tangent formula.

Step 1: Identify the Points of Intersection

Solve the system of equations to find the points where the curves intersect. Set the polar equations equal to each other: \[3 \cos \theta = 1 + \cos \theta\]Rearrange and solve for \theta: \[3 \cos \theta - \cos \theta = 1\]\[2 \cos \theta = 1\]\[\cos \theta = \frac{1}{2}\]Thus, \theta = \frac{\pi}{3} and \theta = \frac{5\pi}{3}. Plug these back into one of the initial equations to find the points in polar coordinates.

Step 2: Compute the Tangent Lines at Intersection Points

Use the derivative \( \frac{dr}{d\theta} \) to find the slope of the tangent lines at each intersection point. For \, r = 3 \cos \theta: \[\frac{dr}{d\theta} = -3 \sin \theta\]For \, r = 1 + \cos \theta: \[\frac{dr}{d\theta} = -\sin \theta\]Evaluate these derivatives at \theta = \frac{\pi}{3} and \theta = \frac{5\pi}{3}.

Step 3: Tangent Slope Substitution

Evaluate the slopes at the points of interest. For \theta = \frac{\pi}{3}: \[\frac{dr}{d\theta} (3 \cos \theta) = -\frac{3 \sqrt{3}}{2}\]\[\frac{dr}{d\theta} (1 + \cos \theta) = -\frac{\sqrt{3}}{2}\]These give the tangent slopes at the intersection points.

Step 4: Calculate the Angle Between the Tangent Lines

The angle between two tangent lines is found using their slopes, m1 and m2, through the formula:\[ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \]Substitute \frac{dr}{d\theta} into the equation for each intersection point. Finally, convert the resulting tangent value into the corresponding angle in radians or degrees.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to represent points on a plane, using a distance and an angle instead of the traditional x and y values. This is particularly useful for describing curves and shapes that have a circular or spiral pattern. In polar coordinates, each point is defined by \((r, \theta)\), where \r\ is the radial distance from the origin, and \theta\ is the angle from the positive x-axis.
For instance, the polar curve represented as \r = 3 \cos \theta\ tells us that the distance varies depending on \theta\. Likewise, \r = 1 + \cos \theta\ describes another curve. To find where these curves intersect, we solve their equations together by setting \r = 3 \cos \theta\ equal to \r = 1 + \cos \theta\.
Through simplification, we find the values of \theta\ where the curves intersect.

Tangent Lines

Tangent lines are straight lines that touch a curve at exactly one point and have the same instantaneous direction as the curve at that point. They are fundamental in understanding the behavior of curves at particular points. To find the tangent line in polar coordinates, we focus on the derivative of the radial distance, \(r\), with respect to the angle, \theta\.
For each curve, we compute the derivative \ \frac{dr}{d\theta} \ to determine the slope of the tangent line at the points of intersection. For example, given \r = 3 \cos \theta, we find its derivative to be \-3 \sin \theta\. Similarly, for \r = 1 + \cos \theta\, the derivative is \-\frac{\sin \theta\.\
This information helps us understand the orientations of the curves at the points where they intersect.

Derivatives

Derivatives measure how a function changes as its input changes. In calculus, the derivative of a function at any point tells us the slope of the tangent line to the function at that point. For polar coordinates, derivatives play a crucial role in determining the slopes of tangent lines to the curves.
To find derivatives of polar functions, we use the chain rule. For example, the derivative of \r = 3 \cos \theta\ with respect to \theta\ is \-3 \sin \theta\. This derivative tells us how the radial distance, \(r\), changes as the angle \theta\ changes. Calculating these derivatives at specific intersection points helps us find the exact slopes of the tangent lines.

Intersection Points

Intersection points are where two curves meet or cross each other. To find these points when dealing with polar coordinates, we set the equations of the curves equal to each other and solve for \theta\.
For the equations \(r = 3 \cos \theta\) and \(r = 1 + \cos \theta\), we set \3 \cos \theta = 1 + \cos \theta\. By simplifying, we find \theta = \frac{\pi}{3}\ and \theta = \frac{5\pi}{3}\. By substituting these \theta\ values back into one of the original polar equations, we can find the specific intersection points. Understanding the coordinates of these intersection points helps in analyzing how curves relate to each other and in calculating angles between their tangent lines.