Problem 15 - 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.\(\left\\{\begin{array}{l}r=\cos \theta \\\ r=\sin 2 \theta\end{array}\right.\)

Step-by-Step Solution

Answer

Calculate intersections: \( \theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6} \).Determine slopes and use \( \tan \phi = \left| \frac{m1 - m2}{1 + m1 m2} \right| \) to find angles.

Step 1: Identify the Curves

The given curves are described by the polar equations: 1) Curve 1: \( r = \cos \theta \)2) Curve 2: \( r = \sin 2 \theta \)

Step 2: Find Points of Intersection

Set the equations equal to find the intersection points: \( \cos \theta = \sin 2 \theta \). Using the double angle identity, \( \sin 2 \theta = 2 \sin \theta \cos \theta \), we get \( \cos \theta = 2 \sin \theta \cos \theta\). Simplifying, we find \( \cos \theta (1 - 2 \sin \theta) = 0 \). Thus, \( \cos \theta = 0 \) or \( \sin \theta = \frac{1}{2} \).

Step 3: Solve for Angles

Solving for \(\theta\): When \( \cos \theta = 0 \), \(\theta = \frac{\pi}{2}, \frac{3\pi}{2} \).When \( \sin \theta = \frac{1}{2} \), \(\theta = \frac{\pi}{6}, \frac{5\pi}{6} \).

Step 4: Convert to Cartesian Coordinates

Convert the polar coordinates to Cartesian coordinates to make differentiation easier: 1) Curve 1: \( x = r \cos \theta = \cos^2 \theta \), \( y = r \sin \theta = \cos \theta \sin \theta \).2) Curve 2: \( x = r \cos \theta = \sin 2\theta \cos \theta \), \( y = \sin 2 \theta \sin \theta = 2 \sin \theta \cos \theta \sin \theta = 2 \sin^2 \theta \cos \theta \).

Step 5: Find the Derivative of Each Curve

Differentiate with respect to \(\theta\): 1) For \( r = \cos \theta \), \(\frac{dy}{dx} = -\tan \theta \).2) For \( r = \sin 2 \theta \), use the chain rule to differentiate and find \(\frac{dy}{dx} \).

Step 6: Calculate the Tangent Lines' Slopes at Intersection Points

Using the intersection points \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{6}, \frac{5\pi}{6}\), calculate the slopes of the tangent lines for each point using the derivatives found in the previous step.

Step 7: Determine the Angle Between Tangent Lines

Use the formula for the angle \(\phi\) between two lines: \( \tan \phi = \left| \frac{m1 - m2}{1 + m1 m2} \right|\), where \( m1 \) and \( m2 \) are the slopes of the tangent lines at the intersection points.

Key Concepts

polar coordinates

Polar coordinates use a different system to describe points in the plane compared to Cartesian coordinates. Instead of using an (x, y) pair, polar coordinates use \(r\) (the distance from the origin) and \(\theta\) (the angle from the positive x-axis).
For example, the point (3, π/4) in polar coordinates represents a point 3 units away from the origin at a 45° angle from the x-axis.
When converting between polar and Cartesian:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

This transformation helps in understanding and solving problems involving curves described by polar equations, such as finding intersection points.

differentiation

Differentiation means finding the rate at which one quantity changes with respect to another. In the context of curves, it helps find the slope of the tangent line at any given point.
For a polar equation \(r(\theta)\), you typically convert to Cartesian coordinates to differentiate with respect to \(\theta\).
For example, given: \( r = \cos \theta \)

• Convert to Cartesian: \( x = \cos^2 \theta, y = \cos \theta \sin \theta \)
• Differentiate: \( \frac{dy}{dx} \)

Using the chain rule helps in more complex situations such as \( r = \sin 2 \theta \), which involves differentiating composite functions.

tangent lines

The tangent line at any point on a curve represents the direction in which the curve is heading at that point. The slope of the tangent line indicates how steep the curve is.
To find the tangent line's slope at a given point:

• First, convert the polar equation to Cartesian coordinates.
• Next, differentiate to find \(\frac{dy}{dx} \).

For the curves given:
1) \(r = \cos \theta\) results in \(\frac{dy}{dx} = -\tan \theta \).
2) \(r = \sin 2 \theta\) requires using the chain rule due to the double angle function.

intersection points

Finding intersection points involves determining where two curves meet. Setting the equations equal helps find common solutions.
In this exercise, setting \(\cos \theta = \sin 2 \theta\) helped find where the curves intersect:

• Using identities, the condition simplifies to \( \cos \theta (1 - 2 \sin \theta) = 0 \).
• Solving: \( \cos \theta = 0 \) leads to \ ( \theta = \frac{\pi}{2}, \frac{3\pi}{2}) \ and \ ( \sin \theta = \frac{1}{2}) \ leads to (\ theta = \frac{\pi}{6}, \frac{5\pi}{6})\.

These angles represent the points of intersection. We then use these points to find the slopes of tangent lines, ultimately calculating the angle between them.