Problem 58 - Calculus Early Transcendentals

Question

\(57-62\) Find the slope of the tangent line to the given polar curve at the point specified by the value of \(\theta\) . $$r=2-\sin \theta, \quad \theta=\pi / 3$$

Step-by-Step Solution

Answer

The slope of the tangent line is \(-\sqrt{3}\).

Step 1: Convert Polar to Cartesian Coordinates

The first step is to convert the polar coordinates given by the equation \( r = 2 - \sin \theta \) to Cartesian coordinates. Using the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \), calculate the Cartesian coordinates as functions of \( \theta \).

Step 2: Differentiate for dy/dx

We need to find the slope of the curve at the given point, so differentiate the expressions for \( x \) and \( y \) with respect to \( \theta \). Use these results to find \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \). Remember, slope \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \).

Step 3: Substitute \( \theta = \pi/3 \) Into Derivatives

Substitute \( \theta = \pi/3 \) into the derivatives \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \) to evaluate these expressions. Calculate the slope \( \frac{dy}{dx} \) of the tangent line using these evaluated derivatives.

Key Concepts

Slope of Tangent Line

The slope of a tangent line to a curve at a particular point gives us valuable information about the curve's behavior and direction at that specific location. Think of it like understanding how steep a hillside is at a certain spot. In polar coordinates, this involves understanding how the angle and radius interact.
To find the slope for a polar curve like the one given by the equation \( r = 2 - \sin \theta \), we follow a methodical process:

• First, express the polar equation in terms of Cartesian coordinates (more on that in the next section).
• Then, differentiate these Cartesian expressions with respect to \( \theta \).
• Finally, evaluate the slope using the formula \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \), especially at the given angle \( \theta = \pi/3 \).

A successful calculation will show how sharply the curve is changing at the point defined by \( \theta = \pi/3 \). This gives a clear view of the tangent's angle relative to the coordinate axes.

Converter Polar to Cartesian

Converting polar coordinates to Cartesian coordinates helps us analyze the curve more effectively, as many calculations and visual interpretations align better with the Cartesian system.
To convert from polar to Cartesian, we utilize the relationships:

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

For the polar curve \( r = 2 - \sin \theta \), substituting \( r \) into these equations gives us expressions for \( x \) and \( y \) which are functions of \( \theta \). Essentially, this transformation allows us to map polar curves onto the familiar \(xy\)-plane by calculating values of \(x\) and \(y\) for any given \(\theta\). This step is crucial because once we have \(x(\theta)\) and \(y(\theta)\), we can proceed to the differentiation step to find the slope of the tangent line.

Differentiation with Respect to Theta

Differentiation with respect to \( \theta \) in polar coordinates is key to finding the slope of the tangent. This means we must differentiate the corresponding Cartesian coordinates \( x(\theta) \) and \( y(\theta) \).
For the equations derived from \( x = r \cos \theta \) and \( y = r \sin \theta \), we need:

• \( \frac{dx}{d\theta} \)
• \( \frac{dy}{d\theta} \)

The derivatives \( \frac{dx}{d\theta} \) and \( \frac{dy}{d\theta} \) describe how the \(x\) and \(y\) values change as the angle \(\theta\) changes. Once these derivatives are computed, the slope of the tangent line at any \(\theta\) can be found via the fraction \( \frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta} \). Substituting \(\theta = \pi/3\) into these derivatives will provide the exact slope of the tangent at that specific angle.

Polar Curves

Polar curves are quite unique because they describe shapes based on a distance from a central point and an angle from a reference direction. Unlike Cartesian coordinates which define a point by \(x\) and \(y\) linear projections, polar coordinates use \( r \) (radial distance) and \( \theta \) (angle).
The given curve \( r = 2 - \sin \theta \) is an example of a polar curve. It represents a path traced out by the radius \( r \), which varies as the angle \( \theta \) progresses. Understanding this type of curve requires understanding how different values of \(\theta\) can change \(r\) and thereby plot out the path of the curve.

• Unlike Cartesian equations, polar equations can produce circles, loops, or other intricate shapes without explicit quadratic or higher degree terms.
• They're often more natural for describing phenomena that naturally involve angles, distances from a point, or periodic processes.

Mastering polar curves opens doors to more advanced topics in mathematics and physics, where such structures are commonly analyzed.