Problem 251 - Calculus Volume 3

Question

Find the points at which the following polar curves have a horizontal or vertical tangent line. $$ r^{2}=4 \cos (2 \theta) $$

Step-by-Step Solution

Answer

Horizontal and vertical tangents occur for specific \(\theta\) values found by setting \(\frac{dy}{d\theta} = 0\) and \(\frac{dx}{d\theta} = 0\), respectively, and using them to find \(r\). Substitute back to find Cartesian coordinates.

Step 1: Understanding Polar and Cartesian Coordinates

To find the tangent lines of a polar curve, we need to convert the polar coordinates into Cartesian coordinates. In polar coordinates, a point is given by \(r, \theta\), which corresponds to Cartesian coordinates \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Since \(r^2 = 4\cos(2\theta)\), first express \(r\) in terms of \(\theta\), and use this to substitute into the Cartesian coordinate formulas.

Step 2: Express r in terms of θ

From the equation \(r^2 = 4\cos(2\theta)\), we get \(r = \pm\sqrt{4\cos(2\theta)} = \pm 2\sqrt{\cos(2\theta)}\). This expression will be used to convert the polar equation into Cartesian form.

Step 3: Differentiating x and y with respect to θ

The Cartesian coordinates are given by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Differentiate these with respect to \(\theta\): \(\frac{dx}{d\theta} = \frac{dr}{d\theta}\cos(\theta) - r\sin(\theta)\) and \(\frac{dy}{d\theta} = \frac{dr}{d\theta}\sin(\theta) + r\cos(\theta)\).

Step 4: Find horizontal tangent lines

For a horizontal tangent line, \(\frac{dy}{d\theta} = 0\) while \(\frac{dx}{d\theta} eq 0\). Substitute \(r \) and \(dr/d\theta\) into the derivative of \(y\), solve \(\frac{dy}{d\theta} = 0\) for \(\theta\). This typically involves solving trigonometric equations.

Step 5: Solve for horizontal tangents using derivatives

Here, using \(r = 2\sqrt{\cos(2\theta)}\), compute \(\frac{dr}{d\theta}\): \(\frac{dr}{d\theta} = -\frac{2\sin(2\theta)}{\sqrt{\cos(2\theta)}}\). Differentiating \(y = r\sin(\theta)\): \(\frac{dy}{d\theta} = -\frac{2\sin(2\theta)}{\sqrt{\cos(2\theta)}}\sin(\theta) + 2\sqrt{\cos(2\theta)}\cos(\theta)\). Set \(\frac{dy}{d\theta} = 0\) to find values for \(\theta\).

Step 6: Find vertical tangent lines

For vertical tangent lines, \(\frac{dx}{d\theta} = 0\) while \(\frac{dy}{d\theta} eq 0\). Substitute \(r \) and \(dr/d\theta\) into the derivative of \(x\), and solve \(\frac{dx}{d\theta} = 0\).

Step 7: Solve for vertical tangents using derivatives

Using the same \(r\) expression and its derivative, we have \(\frac{dx}{d\theta} = -\frac{2\sin(2\theta)}{\sqrt{\cos(2\theta)}}\cos(\theta) - 2\sqrt{\cos(2\theta)}\sin(\theta)\). Solve \(\frac{dx}{d\theta} = 0\) for values of \(\theta\).

Step 8: Determining Points of Tangency

Based on the solutions derived for \(\theta\) in previous steps, substitute these back into the equations for \(r\), \(x\), and \(y\) to determine the Cartesian coordinates (points) where the tangents occur.

Key Concepts

Tangent Lines in Polar and Cartesian Systems

Tangent lines are special lines that touch a curve at just one point without crossing it. These lines indicate the direction in which the curve is heading at that specific point. When dealing with polar coordinates, the process to find tangent lines can seem a bit complex at first. However, once converted to Cartesian coordinates, it becomes easier to calculate where these tangent lines are horizontal or vertical.

In polar coordinates, each point on a plane is determined by a radius (\(r\)) and an angle (\(\theta\)). This system is quite different from the Cartesian system, which uses only the x- and y-coordinates to define a point. When you have a polar curve like \(r^2 = 4\cos(2\theta)\), to work with tangent lines, you first translate these coordinates into the familiar Cartesian system. This involves using the relationships \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). This conversion is crucial as it allows us to apply calculus to find the slope of the curve at any given point.

The slope of the tangent line is defined by the derivatives of \(x\) and \(y\) with respect to \(\theta\). Specifically, horizontal tangent lines occur when the rate of change of \(y\) with respect to \(\theta\) is zero, indicating no vertical movement, while vertical tangent lines occur where the rate of change of \(x\) is zero.

The Role of Cartesian Coordinates

Cartesian coordinates are a mainstay in mathematics because they provide a straightforward way to describe a position in space using two numbers: the x- and y-coordinates. This system is especially powerful when analyzing properties of curves, such as finding tangent lines, by offering an easier basis for calculus operations.

When dealing with polar equations like \(r^2 = 4\cos(2\theta)\), a crucial step is translating these into Cartesian coordinates using the formulas \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). By expressing \(r\) in terms of \(\theta\), one can substitute back into these expressions to find the Cartesian equivalents. So, \(r = \pm 2\sqrt{\cos(2\theta)}\) allows you to substitute back to get \(x\) and \(y\) in terms of \(\theta\).

• Helps simplify calculations by transforming complex polar curves into a more manageable form.
• Allows for the application of derivatives to find slopes of tangent lines.

By deriving functions for \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\), you can set them to zero to detect where the changes in x or y happen respectively, which is key in identifying the exact points of tangency on the curve.

Trigonometric Equations in Finding Tangents

Trigonometric equations are pivotal when handling problems involving polar coordinates since polar equations often involve trigonometric functions. In this polar curve example, the equation \(r^2 = 4\cos(2\theta)\) shows how \(r\) is dependent on the cosine function, a classic trigonometric function.

To identify the points where tangent lines are horizontal or vertical, trigonometric identities are essential. These identities help transform, simplify, and solve these equations and expressions, making it feasible to find derivatives and setting conditions to zero for solving tangent lines.

• Horizontal tangents arise when \(\frac{dy}{d\theta} = 0\) while \(\frac{dx}{d\theta} eq 0\).
• Vertical tangents occur when \(\frac{dx}{d\theta} = 0\) and \(\frac{dy}{d\theta} eq 0\).

In this context, differentiating the expressions \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) produces the necessary derivatives for solving these conditions. Solving these derivative-set equations often involves dealing with trigonometric identities, like the double angle formula \(\cos(2\theta)\), ensuring you can solve for \(\theta\) and determine precise tangent points on the curve.