Problem 92 - Essential Calculus

Question

Use the result of Exercise 90 to find the angle \(\psi\) between the radial and tangent lines to the graph for the indicated value of \(\theta\). Use a graphing utility to graph the polar equation, the radial line, and the tangent line for the indicated value of \(\theta .\) Identify the angle \(\psi\). $$ r=4 \sin 2 \theta, \theta=\pi / 6 $$

Step-by-Step Solution

Answer

The angle \(\psi\) between the radial and tangent lines to the graph is 60 degrees.

Step 1 - Compute the derivative of r

First, determine the derivative of \(r\) with respect to \(\theta\). The derivative of \(r = 4 \sin 2\theta\) with respect to \(\theta\) can be calculated as \(dr/d\theta = 8 \cos 2\theta\).

Step 2 - Substitute \(\theta = \pi/6\)

Secondly, substitute the given value of \(\theta = \pi/6\). Observe that at \(\theta = \pi/6\), \(r = 4 \sin(2 \cdot \pi/6) = 4 \sin(\pi/3) = 4 \cdot \sqrt{3}/2 = 2\sqrt{3}\) and \(dr/d\theta = 8 \cos(2 \cdot \pi/6) = 8 \cos(\pi/3) = 8 \cdot 1/2 = 4\).

Step 3 - Find the slope of the tangent line

Next, determine the slope of the tangent line from the formula \(m = -r/\(dr/d\theta)\), which means \(m = -2\sqrt{3}/4 = -\sqrt{3}/2\).

Step 4 - Determine the angle \(\psi\)

Finally, recall that the angle between the x-axis and the tangent line is given by \(arctan(m)\). Hence the angle \(\phi\) is \(180° - arctan(m)\), which is \(180° - arctan(-\sqrt{3}/2) = 180° - 120° = 60°\). Therefore, the angle \(\psi\) between the radial and tangent lines to the graph is \(60°\).

Key Concepts

Polar Coordinates

Polar coordinates provide a different way of representing points on a plane compared to the more common Cartesian coordinates. In the polar system, each point is determined by a distance from a fixed origin (called the radial coordinate \( r \)) and an angle \( \theta \) relative to a fixed direction (usually the positive x-axis).

This can be particularly useful in scenarios involving curves and circular shapes, as polar coordinates can simplify the representation of equations and make certain calculations easier to perform. Unlike the Cartesian system that uses \((x, y)\), polar coordinates use \((r, \theta)\):

• \( r \): Measures how far away the point is from the origin.
• \( \theta \): Represents the angle formed with the positive x-axis.

In many mathematical problems, using polar coordinates can simplify the process of finding lengths, angles, and slopes by reducing complex algebraic expressions to trigonometric ones. This approach is particularly handy when dealing with graphs of equations that describe curves, such as spiral or elliptical paths.

Derivatives in Polar Form

The concept of derivatives extends to polar coordinates and is crucial when analyzing how a polar curve behaves. The derivative of \( r \) with respect to \( \theta \), denoted as \( dr/d\theta \), represents the rate at which the radial distance \( r \) changes as the angle \( \theta \) increases.

In a polar function of the form \( r = f(\theta) \), calculating \( dr/d\theta \) gives insight into the growth or shrinkage of the radius as one moves along the curve. For example, in the exercise provided, the function \( r = 4 \sin 2\theta \) has a derivative \( dr/d\theta = 8 \cos 2\theta \). Substituting \( \theta = \pi/6 \) into this derivative helps us understand how quickly the radius is changing at that specific angle.

• This derivative is used to find the slope of the tangent line.
• It illustrates the dynamic nature of the curve as \( \theta \) varies.

Understanding derivatives in polar form is essential for applications such as determining the slope of tangent lines or analyzing the behavior of radial distances for changing angles. It connects the geometric properties of curves to calculus and enhances problem-solving abilities related to angles and slopes.

Slope of Tangent Line

The slope of a tangent line in polar coordinates can be found using the derivative of \( r \) with respect to \( \theta \). This slope tells us how steep the tangent line is at any given point on the curve.

For a polar curve described by \( r = f(\theta) \), the slope \( m \) of the tangent line is calculated using the relationship \( m = -r / (dr/d\theta) \). This formula considers both the radial distance \( r \) and the rate of change \( dr/d\theta \). In the provided exercise, substituting \( r = 2\sqrt{3} \) and \( dr/d\theta = 4 \) gives us a slope \( m = -\sqrt{3}/2 \).

This step is fundamental in understanding not just how steep a line is, but how the curve behaves at specific points. The slope facilitates easier analysis and helps in plotting or estimating tangent lines, which act as linear approximations of the curve nearby.

• A positive slope indicates the tangent line is rising.
• A negative slope suggests the tangent line is falling.
• A zero slope results in a horizontal line.

The angle \( \psi \) between the radial and tangent lines can also be deduced from this slope. By finding \( \arctan(m) \), we can further explore relationships between curves and angles, providing insight into the geometric properties encountered in polar systems.