Problem 40 - Calculus Early Transcendentals

Question

\(37-42\) Find all points of intersection of the given curves. $$ r=\cos 3 \theta, \quad r=\sin 3 \theta $$

Step-by-Step Solution

Answer

The curves intersect at points like \(\left(\frac{\sqrt{2}}{2}, \frac{\pi}{12} \right)\) and \(\left(-\frac{\sqrt{2}}{2}, \frac{5\pi}{12} \right)\).

Step 1: Set the Equations Equal

To find the points of intersection of the given polar curves, set them equal to each other: \( \cos 3\theta = \sin 3\theta \).

Step 2: Use Trigonometric Identity

We know that \( \cos 3\theta = \sin 3\theta \) implies \( \tan 3\theta = 1 \). This is because \( \tan 3\theta = \frac{\sin 3\theta}{\cos 3\theta} \).

Step 3: Solve for \(\theta\)

The equation \( \tan 3\theta = 1 \) is true when \( 3\theta = \frac{\pi}{4} + k\pi \), where \( k \) is an integer. Divide by 3 to solve for \( \theta \): \( \theta = \frac{\pi}{12} + \frac{k\pi}{3} \).

Step 4: Determine the Points of Intersection

Substitute each of the values of \(\theta\) from Step 3 back into either original equation (e.g., \( r = \cos 3\theta \) or \( r = \sin 3\theta \)) to find the corresponding \( r \) values. For each \(\theta\), compute \( r = \cos\big(\frac{\pi}{4} + k\pi\big) \). Evaluating a few values, you get points \(\left(\frac{\sqrt{2}}{2}, \frac{\pi}{12} \right)\), \(\left(-\frac{\sqrt{2}}{2}, \frac{5\pi}{12} \right)\), etc.

Key Concepts

Polar Coordinates

Polar coordinates represent a different system of locating points based on their distance from the origin and angle from a fixed direction. Instead of the typical Cartesian coordinates (x, y), polar coordinates are defined as (r, θ), where:

• r is the radial distance from the origin.
• θ is the angle in radians from the positive x-axis.

This system is particularly useful for dealing with curves and circular shapes. For this exercise, we're looking at curves defined by the equations r = cos 3θ and r = sin 3θ.
These are polar equations representing curves that extend radially outward depending on the angle θ. The challenge lies in finding where these curves intersect, which requires understanding how changes in θ affect the value of r for both expressions.

Trigonometric Identities

Trigonometric identities are essential tools for solving equations, particularly when dealing with polar curves. They allow us to rewrite expressions to find solutions more easily.
In our example, the identity \[tan 3θ = \frac{sin 3θ}{cos 3θ}\]helps us by transforming an equality between sin and cos into one involving the tangent function. Since \[\tan 3θ = 1\]implies a 45-degree angle (or equivalent), we can find potential solutions for θ that satisfy the initial condition.
Understanding trigonometric identities and transformations can simplify solving intersections of polar curves by allowing us to manipulate equations into more usable forms.

Solving Equations

Solving equations derived from the values of polar curves often involves several steps. We begin by equating the curves:
\[cos 3θ = sin 3θ\]Through the identity transformation discussed earlier, this converts to\[\tan 3θ = 1\]From this, we identify solutions using the arctangent function, leading to multiple angles θ:
\[3θ = \frac{π}{4} + kπ\]where k is an integer. Solving for θ gives:\[θ = \frac{π}{12} + \frac{kπ}{3}\]This yields several values, each corresponding to a point of intersection.
Finally, to find the specific polar coordinates (r, θ):

• Substitute each θ back into the original equations to solve for r.
• Using examples, such as θ = \(\frac{π}{12}\), calculate r from \(cos 3θ\) or \(sin 3θ\).

This method unveils the intersection points, showcasing the elegance of using both trigonometric identities and polar transformations to uncover solutions.