Question

For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. $$r=3+\cos (2\theta ),\theta =\frac{3\pi }{4}$$

Short answer

Tangent line: $y=\frac{3\sqrt{2}}{2}$.

Step-by-step solution

Find the Derivative of the Polar Equation

The given polar equation is $r=3+\cos (2\theta )$. To find the derivative of $r$ with respect to $\theta$, apply the chain rule. The derivative $\frac{dr}{d\theta }=-2\sin (2\theta )$.

Evaluate the Radius and Derivative at $\theta =\frac{3\pi }{4}$

Substitute $\theta =\frac{3\pi }{4}$ into the polar equation to find the radius: $r=3+\cos (2\cdot \frac{3\pi }{4})=3+\cos (\frac{3\pi }{2})=3$. Evaluate the derivative: ${\frac{dr}{d\theta }|}_{\theta =\frac{3\pi }{4}}=-2\sin (\frac{3\pi }{2})=0$.

Convert Polar Coordinates to Rectangular Coordinates

With $\theta =\frac{3\pi }{4}$ and $r=3$, the rectangular coordinates $(x,y)$ are given by: $x=r\cos (\theta )=3\cos (\frac{3\pi }{4})=-\frac{3\sqrt{2}}{2}$ and $y=r\sin (\theta )=3\sin (\frac{3\pi }{4})=\frac{3\sqrt{2}}{2}$.

Calculate the Slope of the Tangent Line

The slope of the tangent line at a point in polar coordinates can be determined as $\frac{dy}{d\theta }/\frac{dx}{d\theta }$ with respect to $\theta$. Using $x=r\cos (\theta )$ and $y=r\sin (\theta )$, calculate $\frac{dx}{d\theta }$ and $\frac{dy}{d\theta }$ at $\theta =\frac{3\pi }{4}$: $\frac{dx}{d\theta }=-3\sqrt{2}$ and $\frac{dy}{d\theta }=0$. Thus, the slope $m=\frac{0}{-3\sqrt{2}}=0$.

Write the Equation of the Tangent Line

Using point-slope form $y-y_1=m(x-x_1)$, where $(x_1,y_1)=(-\frac{3\sqrt{2}}{2},\frac{3\sqrt{2}}{2})$ and $m=0$, the equation of the tangent line is $y=\frac{3\sqrt{2}}{2}$.

Graph Representation

Plot the function $r=3+\cos (2\theta )$ in polar coordinates as a closed curve. The tangent line, a horizontal line, intersects at $x=-\frac{3\sqrt{2}}{2}$, repeating this for various $\theta$ values visually verifies the solution.

Key concepts

Understanding the Tangent Line

A tangent line is a straight line that just "touches" a curve at a specific point without crossing it, resembling the way a pencil might rest softly against a piece of paper. In essence, it represents the curve’s direction or slope at that exact point. When you imagine a curve drawn on a piece of paper, a tangent line sits at a particular spot, forming an angle without intersecting the curve fully.
To find the equation of a tangent line in a polar coordinate setup like the one given, we aim to evaluate how the curve behaves at a given angle. In this exercise, the angle is denoted as $\theta =\frac{3\pi }{4}$. We ascertain that at this point, the slope of the tangent line is zero. This intuition that the slope of a tangent might sometimes be zero is crucial as it identifies areas where the curve might be momentarily flat or the direction does not change.

How Derivatives Help Us

Derivatives are incredibly dynamic tools in mathematics. They allow us to examine how functions change and help in finding slopes of lines tangent to curves. In polar coordinates—where positions are described using radius and angle—derivatives depict how the radius changes as the angle changes. Consider the polar equation $r=3+\cos (2\theta )$. By taking its derivative $\frac{dr}{d\theta }=-2\sin (2\theta )$, we can predict the curve's behavior relative to angular changes.
By finding the value of the derivative at $\theta =\frac{3\pi }{4}$, namely $0$, we understand that there is no change in the behavior of $r$ at that angle. Meaning, the tangent here aligns horizontally, denoting either a peak or trough on the curve. Recognizing this tells us that there are points on curves that might appear relatively flat when examining their slope through derivatives.

Switching to Rectangular Coordinates

Although the original problem is set up in polar coordinates, mathematicians often translate these into rectangular coordinates for certain types of analyses. This is because rectangular coordinates are often more intuitive—they describe a point by how far along and how far up or down it is on a grid-like layout.
In this task, we transform the polar coordinates $(r,\theta )=(3,\frac{3\pi }{4})$ into rectangular coordinates $(x,y)$. This is done using the equations:

• $x=r\cos (\theta )=3\cos (\frac{3\pi }{4})=-\frac{3\sqrt{2}}{2}$
• $y=r\sin (\theta )=3\sin (\frac{3\pi }{4})=\frac{3\sqrt{2}}{2}$

These calculations position the point accurately within the familiar grid of the X-Y plane, aiding us in further exploring its tangent line. Understanding the shift into rectangular coordinates here ensures broader applicability of skills across different mathematical problems, as it is a frequently used technique in more complex calculus environments.