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Chapter 1: Problem 156

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$r=\sec \theta$$

Short Answer

Expert verified
The graph is a vertical line at \( x = 1 \).

Step by step solution

01

Understand the Polar Equation

The given polar equation is \( r = \sec \theta \). In polar coordinates, \( r \) represents the radial distance from the origin, and \( \theta \) represents the angle measured counterclockwise from the positive x-axis. The secant function is the reciprocal of the cosine function, which means \( \sec \theta = \frac{1}{\cos \theta} \).
02

Express r in Terms of Trigonometric Functions

Let us express \( r \) using cosine: \( r = \frac{1}{\cos \theta} \). This can be rewritten as \( r \cos \theta = 1 \).
03

Convert to Rectangular Coordinates

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). From Step 2, we have \( r \cos \theta = 1 \), which in rectangular coordinates becomes \( x = 1 \).
04

Interpret the Rectangular Equation

The equation \( x = 1 \) is a vertical line in the Cartesian coordinate plane. This means the polar equation \( r = \sec \theta \) describes a vertical line at \( x = 1 \).
05

Graph the Polar Equation

The graph of the polar equation \( r = \sec \theta \) will be a vertical line where each point on the line has an x-coordinate of 1. This confirms the conversion to rectangular coordinates, as any point on the line at \( x = 1 \) satisfies the original polar equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a straightforward way to pinpoint any location in a plane. They're defined by two numbers:
  • \( x \): the horizontal position on the plane.
  • \( y \): the vertical position on the plane.
Together, these coordinates are represented as \((x, y)\). Each point in a two-dimensional space falls exactly at one unique pair of these values. A simple way to visualize this is by picturing a grid where each intersection has a specific \( x \) and \( y \) value. In the exercise, converting the polar equation to the rectangular equation resulted in the
expression \( x = 1 \). This describes a vertical line on the Cartesian plane, where every point has an \( x \)-coordinate of 1 regardless of the \( y \)-value.
Polar Equations
Polar equations provide a unique way to describe curves in a plane through radial distance and angular measurement. The polar coordinates are determined by:
  • \( r \): the radial distance from the origin.
  • \( \theta \): the angle from the positive x-axis.
These equations may initially appear complex but offer a powerful means to encode intricate curves, such as spirals and circles, using functional equations. The polar equation \( r = \sec \theta \) translates the relationship in terms of the angle, where the radial distance changes as the angle changes. Each point on the curve is defined by an \( r \)-\( \theta \) pair, providing a different perspective from the standard Cartesian method.
Trigonometric Functions
Trigonometric functions are crucial when working with polar coordinates as they connect angle measurements with distances, simplifying various mathematical conversions.
  • \( \cos \theta \): relates the adjacent side over hypotenuse in a right triangle.
  • \( \sec \theta \): is the reciprocal of \( \cos \theta \), so \( \sec \theta = \frac{1}{\cos \theta} \).
These relationships are pivotal in converting polar equations to rectangular equations. In this context, expressing \( r = \sec \theta \) as \( r \cos \theta = 1 \) simplifies the process, bridging the gap between the polar and rectangular coordinate systems, ultimately leading to the rectangular equation \( x = 1 \). This reveals the much easier-to-interpret line when viewed through Cartesian geometry.
Graphing
Graphing is a fundamental tool to visualize equations, be they in polar or rectangular form. For effective graphing:
  • Understand the equations you want to plot.
  • Determine whether to plot in polar coordinates (curves, circles) or Cartesian coordinates (straight lines, simple parabolas).
Polar graphs can look markedly different from their rectangular counterparts, offering a unique peek into the behavior of equations as you adjust \( \theta \). In this exercise, the graph of \( r = \sec \theta \) equates to a vertical line \( x = 1 \) in the rectangular coordinate system, showcasing uniform distance from the y-axis and consistent \( x \)-values. Mastering the transition and graphing ensures a deeper grasp of how mathematical representations interconnect.

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Most popular questions from this chapter

For the following exercises, find the slope of a tangent line to a polar curve \(r=f(\theta) .\) Let \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta, \quad\) so the polar equation \(r=f(\theta)\) is now written in parametric form. $$ r=1-\sin \theta ;\left(\frac{1}{2}, \frac{\pi}{6}\right) $$

There is a curve known as the "Black Hole." Use technology to plot \(r=e^{-0.01 \theta}\) for \(-100 \leq \theta \leq 100\)

For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$r=\csc \theta$$

For the following equations, determine which of the conic sections is described. \(52 x^{2}-72 x y+73 y^{2}+40 x+30 y-75=0\)

Determine the Cartesian equation describing the orbit of Pluto, the most eccentric orbit around the Sun. The length of the major axis is 39.26 AU and minor axis is 38.07 AU. What is the eccentricity?
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