Problem 250 - Calculus Volume 3

Question

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ r=4 \cos \theta $$

Step-by-Step Solution

Answer

Horizontal tangents: \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}\). Vertical tangents: \(\theta = 0, \frac{\pi}{2}\).

Step 1: Understand Polar Coordinates

In polar coordinates, a point is determined by the distance from the origin, \(r\), and the angle from the positive x-axis, \(\theta\). The curve given is \(r = 4\cos \theta\). We need to find where this curve has horizontal or vertical tangents.

Step 2: Convert to Cartesian Coordinates

For the purposes of finding tangents, we convert the polar equation to Cartesian coordinates. Recall the conversion formulas: \(x = r \cos\theta\) and \(y = r \sin\theta\). Substitute \(r = 4 \cos \theta\) into these equations:- \(x = (4 \cos \theta) \cos \theta = 4 \cos^2 \theta\)- \(y = (4 \cos \theta) \sin \theta = 4 \cos \theta \sin \theta\).

Step 3: Express \(y\) and \(x\) in terms of \(r\)

Using identities, we know \(\cos \theta = \frac{x}{r}\) and \(\sin \theta = \frac{y}{r}\). In this problem, \(r = 4 \cos \theta\), we have:- \(x = 4\cos^2\theta = r \cos\theta\)- \(y = 4\cos\theta\sin\theta = r \sin\theta\).

Step 4: Find Derivatives and Conditions

To find horizontal tangents, we need \(\frac{dy}{d\theta} = 0\), and to find vertical tangents, we need \(\frac{dx}{d\theta} = 0\).Using derivative rules:- For \(x = 4\cos^2\theta\), you have \(\frac{dx}{d\theta} = -8 \cos \theta \sin \theta\)- For \(y = 4\cos\theta\sin\theta\), you have \(\frac{dy}{d\theta} = 4(\cos^2\theta - \sin^2\theta)\).

Step 5: Solve for Horizontal Tangents

Set \(\frac{dy}{d\theta} = 0\):\[4(\cos^2\theta - \sin^2\theta) = 0\]\[\cos 2\theta = 0\]Solving gives \(2\theta = \frac{\pi}{2}, \frac{3\pi}{2}, ...\) hence \(\theta = \frac{\pi}{4}, \frac{3\pi}{4}, ...\).Substitute back to polar to find points where these occurs, solving for \(r\).

Step 6: Solve for Vertical Tangents

Set \(\frac{dx}{d\theta} = 0\):\[-8 \cos\theta \sin\theta = 0\]This occurs when \(\sin\theta = 0\) or \(\cos\theta = 0\), which simplifies to \(\theta = 0, \pi/2, \pi, 3\pi/2, ...\).Compute corresponding \(r\) for these \(\theta\) values.

Key Concepts

Horizontal Tangent

A horizontal tangent line on a curve implies that at this particular point, the change in the vertical direction is zero, while there may still be change in the horizontal direction. This means the slope of the tangent line is equal to zero. In polar coordinates, when we want to find where a curve like \( r = 4 \cos \theta \) has a horizontal tangent, we examine the derivative \( \frac{dy}{d\theta} \).

• If \( \frac{dy}{d\theta} = 0 \), the tangent is horizontal.

In our exercise, we found this condition by solving the equation \( 4(\cos^2 \theta - \sin^2 \theta) = 0 \), simplifying it gives \( \cos 2\theta = 0 \). To find the exact angles where this occurs, we solve for \( \theta \) which gives us \( \theta = \frac{\pi}{4}, \frac{3\pi}{4}, ...\). Since \( \theta \) represents the angle in polar coordinates, it is crucial to evaluate \( r \) to determine the actual points on the curve.

Vertical Tangent

A vertical tangent line occurs when the change in the horizontal direction is zero at a particular point on the curve. To find these points in polar coordinates, we look for where the derivative \( \frac{dx}{d\theta} \) is zero.

• A \( \frac{dx}{d\theta} = 0 \) indicates a vertical tangent.

For the polar equation \( r = 4 \cos \theta \), the task was to find where the vertical tangent occurs. We calculate \( \frac{dx}{d\theta} = -8 \cos \theta \sin \theta \) and set it to zero. Solving \( -8 \cos \theta \sin \theta = 0 \), we find \( \sin \theta = 0 \) or \( \cos \theta = 0 \). These conditions give the angles \( \theta = 0, \pi/2, \pi, 3\pi/2, ... \). At these angles, the values of \( r \) help identify the curve's relevant points.

Polar Equation

A polar equation is a way of defining a curve using polar coordinates, which allows you to represent the location of a point based on its distance from the origin (called radius \( r \)) and the angle \( \theta \) relative to a fixed direction, usually the positive x-axis. The equation \( r = 4 \cos \theta \) is a classic example of a polar equation that represents a circle. Here, \( r \) changes with \( \theta \) in such a way that creates symmetry. Transforming this equation into Cartesian coordinates helps in analyzing properties like tangents, which are pivotal in understanding the behavior of polar curves. With the aid of polar equations, complex curves can be described succinctly.

Cartesian Coordinates

Cartesian coordinates provide a system where a point on a plane is designated by its distance along the x-axis (horizontal) and y-axis (vertical), unlike polar coordinates, which use radius and angle. Translating a polar equation into Cartesian coordinates involves using:

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

For the curve \( r = 4 \cos \theta \), conversion involves expressing both \( x \) and \( y \) in terms of \( \cos \theta \), resulting eventually into equations for \( x \) and \( y \) which are easier to differentiate. Differentiation of these functions detects points of interest, namely horizontal and vertical tangents, clarifying the geometric nature of polar curves in familiar Cartesian terms. The translation lets us visualize and work with polar curves within the traditional Cartesian framework we are accustomed to.