Problem 66 - Thomas Calculus

Question

Graph the lines and conic sections in Exercises \(65-74.\) $$r=4 \sec (\theta+\pi / 6)$$

Step-by-Step Solution

Answer

The graph is a straight line with a slope of \( \sqrt{3} \) and y-intercept of -8.

Step 1: Understanding the Given Equation

The given polar equation is \( r = 4 \sec(\theta + \pi/6) \). \( \sec \) is the secant function, which in cartesian coordinates is the reciprocal of \( \cos \). In polar coordinates, the relationship between \( r \), \( \theta \), and Cartesian coordinates \( x \) and \( y \) are expressed as \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Transforming to Cartesian Coordinates

To graph \( r = 4 \sec(\theta + \pi/6) \), we first convert the equation to Cartesian coordinates. Since \( \sec \theta = \frac{1}{\cos \theta} \), the equation can be rewritten as \( r \cos(\theta + \pi/6) = 4 \). Noting that \( \cos(\theta + \pi/6) = \cos \theta \cos \frac{\pi}{6} - \sin \theta \sin \frac{\pi}{6} \), substitute \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Step 3: Simplifying the Equation

The equation becomes \( r(\cos \theta \cdot \frac{\sqrt{3}}{2} - \sin \theta \cdot \frac{1}{2}) = 4 \). Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into this equation to get \( (\frac{\sqrt{3}}{2})x - (\frac{1}{2})y = 4 \).

Step 4: Rearranging for the Line Equation

Multiply the entire equation by 2 to clear the fractions: \( \sqrt{3}x - y = 8 \). This is the Cartesian equation of a straight line.

Step 5: Identifying the Slope and Y-intercept

The equation \( \sqrt{3}x - y = 8 \) is in standard form. To find the slope and y-intercept, rearrange it to slope-intercept form \( y = mx + c \) where \( m \) is the slope. The equation becomes \( y = \sqrt{3}x - 8 \). The slope \( m \) is \( \sqrt{3} \) and the y-intercept is \( -8 \).

Step 6: Sketching the Graph

On the graph, plot the y-intercept at -8 on the y-axis. Use the slope \( \sqrt{3} \approx 1.732 \) which means for every unit increase in x, y increases by approximately 1.732 units. Draw the line through this point using the slope.

Key Concepts

Polar Coordinates

Polar coordinates offer a way to describe the position of points on a plane. Instead of using horizontal and vertical measurements like Cartesian coordinates, polar coordinates use a combination of a radius and an angle.

• The radius, denoted as \( r \), is the distance from the origin to the point.
• The angle, \( \theta \), is measured from a reference direction, typically the positive x-axis, in a counter-clockwise direction.

The equation given in the exercise is in polar form: \( r = 4 \sec(\theta + \pi/6) \). Here, the value of \( \theta \) is adjusted to account for a phase shift, ensuring that the behavior of \( r \) remains consistent across different angles. This unique angle adjustment differentiates polar equations from their Cartesian counterparts.

Secant Function

The secant function, written as \( \sec(\theta) \), is intimately related to the more commonly known cosine function. It is actually the reciprocal:

• \( \sec(\theta) = \frac{1}{\cos(\theta)} \)

This function is useful in polar equations because it helps in representing relationships involving circles and waves.

When it is used in transformations like the one in our exercise, the secant directly affects the amplitude or distance from the origin. Given in the problem, the secant function shift by \(-\pi/6\) alters the radius, changing how the graph spreads around the origin, essential for moving polar equations into the Cartesian plane.

Cartesian Coordinates

Cartesian coordinates are perhaps the most familiar system for plotting points. They use two values, \( x \) and \( y \), to describe a location on the plane:

• \( x \) is the horizontal distance from the origin.
• \( y \) is the vertical distance from the origin.

To convert from polar to Cartesian, we use these formulas:

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

In the problem, converting \( r = 4 \sec(\theta + \pi/6) \) to Cartesian form was crucial. By expressing \( r \) in terms of \( x \) and \( y \), we transform the polar equation into a linear equation, which is "\( \sqrt{3}x - y = 8 \)". This allows us to plot and analyze the function on a standard Cartesian graph.

Equation of a Line

The equation of a line in Cartesian coordinates can take multiple forms, but the slope-intercept form and standard form are the most common.

• The standard form is \( Ax + By = C \).
• The slope-intercept form is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.

By rearranging \( \sqrt{3}x - y = 8 \), we derive the slope-intercept form, \( y = \sqrt{3}x - 8 \).

• In this equation, the slope \( \sqrt{3} \approx 1.732 \), indicates the steepness of the line.
• The y-intercept at \( -8 \) tells us where the line crosses the y-axis.

These two parameters provide a complete picture of the line, allowing us to sketch it accurately on a graph.