Question

Find the points at which the following polar curves have a horizontal or vertical tangent line.Show that the curve \(r=\sin \theta \tan \theta\) (called a cissoid of Diocles) has the line \(x=1\) as a vertical asymptote.

Short answer

The cissoid has vertical asymptote at \(x=1\); tangent points need \(\frac{dy}{dx}\) analysis.

Step-by-step solution

Convert Polar to Cartesian Coordinates

The polar equation given is \( r = \sin \theta \tan \theta \). To find horizontal and vertical tangents in terms of Cartesian coordinates \((x, y)\), first convert the expression. The relationships between polar and Cartesian coordinates are: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( r \) into these equations:\[ x = \sin \theta \tan \theta \cos \theta \]\[ y = \sin^2 \theta \tan \theta \]

Find Derivative of Cartesian Representation

To find horizontal and vertical tangents, consider \( \frac{dy}{dx} \). Using implicit differentiation, the chain rule gives:\[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]Calculate derivatives:- \( \frac{dy}{d\theta} = 2\sin \theta \cos \theta \tan \theta + \sin^2 \theta (\sec^2 \theta) \)- \( \frac{dx}{d\theta} = \cos \theta \sin \theta \sec^2 \theta - \sin \theta \tan \theta \sin \theta \)Then substitute these into the ratio \( \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \).

Identify the Horizontal and Vertical Tangents

For horizontal tangents, set \( \frac{dy}{dx} = 0 \), which occurs when \( \frac{dy}{d\theta} = 0 \) and \( \frac{dx}{d\theta} eq 0 \). For vertical tangents, set \( \frac{dx}{d\theta} = 0 \), which occurs when \( \frac{dy}{d\theta} eq 0 \). Solve these equations, considering trigonometric identities and possible values of \( \theta \).

Determine Vertical Asymptote

To show \( x = 1 \) is a vertical asymptote, consider where the expression \( x \to 1 \). Note from the equation \( x = \sin \theta \tan \theta \cos \theta = 1 \). This implies that as \( \theta \to 0 \), \( x \to 1 \). Showing that \( x \) approaches the value 1 from the left or right leads us to conclude \( x = 1 \) is a vertical asymptote.

Key concepts

Horizontal Tangent

Understanding horizontal tangents in polar curves involves finding when the slope of the tangent line is zero. For Cartesian coordinates, the derivative \( \frac{dy}{dx} \) represents this slope. If this derivative equals zero, the curve has a horizontal tangent.

In our problem, after converting the polar curve to Cartesian form, the derivative of \( y \) with respect to \( x \), \( \frac{dy}{dx} \), is calculated using the chain rule. Setting \( \frac{dy}{dx} = 0 \) isolates the cases where the tangent is horizontal. This typically involves solving \( \frac{dy}{d\theta} = 0 \) and ensuring \( \frac{dx}{d\theta} eq 0 \).

Solving these trigonometric equations helps us pinpoint specific values of \( \theta \) where the polar curve exhibits horizontal tangents. Knowing these points is essential for fully analyzing the curve's graphical behavior.

Vertical Tangent

Vertical tangents are identified by setting up conditions where the slope \( \frac{dy}{dx} \) becomes undefined or the limit approaches infinity. This happens when \( \frac{dx}{d\theta} = 0 \), provided \( \frac{dy}{d\theta} eq 0 \).

In our given exercise, after converting the polar equation \( r = \sin \theta \tan \theta \) to Cartesian form, we look at the derivatives \( \frac{dy}{d\theta} \) and \( \frac{dx}{d\theta} \). Setting \( \frac{dx}{d\theta} = 0 \) gives conditions for vertical tangency due to an undefined slope.

By solving these equations, we find the exact points or angles at which the curve has vertical tangents. This is crucial for comprehensively understanding the curve's structure, particularly where it sharply moves upwards or downwards.

Polar to Cartesian Conversion

Converting between polar and Cartesian coordinates is a necessary step in analyzing polar curves, especially when dealing with tangents. Polar coordinates describe a point in terms of its distance from the origin \( r \) and the angle \( \theta \), whereas Cartesian relies on \( x \) and \( y \).

In the problem for \( r = \sin \theta \tan \theta \), we use the conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \). This translates the curve into the familiar Cartesian system \( (x, y) \).

Using these conversions, the problem becomes manageable with conventional calculus techniques to find horizontal and vertical tangents. Understanding how to switch between systems aids in visualizing and solving complex problems related to polar curves.

Vertical Asymptote

In curve analysis, identifying a vertical asymptote informs us about a behavior where a curve infinitely approaches a vertical line but never actually touches it. For the curve \( r = \sin \theta \tan \theta \), showing \( x = 1 \) as a vertical asymptote requires recognition of how the curve behaves as it approaches this line.

Analyzing the Cartesian form \( x = \sin \theta \tan \theta \cos \theta \), we examine where \( x \to 1 \). As \( \theta \to 0 \), the expression for \( x \) approaches 1. Whether approaching from the left or right, \( x \) nearing the value 1 confirms this vertical asymptote.

Vertical asymptotes highlight important structural elements of curves, often indicating lines that the function values cannot cross or touch, hence a boundary behavior.