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Chapter 7: Problem 89

For each rectangular equation, write an equivalent polar equation. $$x=4$$

Short Answer

Expert verified
The polar equation equivalent to \( x = 4 \) is \( r \cos(\theta) = 4 \).

Step by step solution

01

Understand the Rectangular Equation

The given rectangular equation is \(x = 4\). This represents a vertical line in the Cartesian coordinate system at \(x = 4\).
02

Know the Polar Coordinates

In polar coordinates, the location of a point is given by \( (r, \theta) \), where \( r \) is the radius (distance from the origin) and \( \theta \) is the angle from the positive x-axis.
03

Convert the x-coordinate

In polar coordinates, \( x = r \cos(\theta) \). Replace \( x \) with this equivalent expression in the given equation: \( r \cos(\theta) = 4 \).
04

Simplify the Expression

The equation \( r \cos(\theta) = 4 \) is the equivalent polar form of the given rectangular equation \( x = 4 \).

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Key Concepts

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

Polar Coordinates
In the polar coordinate system, points are determined by their distance from the origin and the angle they make with the positive x-axis. These coordinates are represented as \( (r, \theta) \).
  • \(r\): This is the radius or the distance from the origin (point (0,0)).
  • \(\theta\): This is the angle measured in radians or degrees from the positive x-axis.

To visualize, imagine starting from the origin and moving outward along a line. The length of this line is \(r\), and the direction of this line relative to the positive x-axis is \(\theta\). Together, these define a point in the polar coordinate system.
For instance, a point with polar coordinates \((5, \frac{\pi}{4})\) is located 5 units away from the origin at an angle of \(\frac{\pi}{4}\) from the positive x-axis.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, define a point's position in a plane using two perpendicular axes: the x-axis and the y-axis. These coordinates are represented as \((x, y)\).
  • \(x\): The horizontal distance from the origin along the x-axis.
  • \(y\): The vertical distance from the origin along the y-axis.

In other words, rectangular coordinates describe how far left/right and up/down a point is from the origin. For example, in the coordinate pair \((3, 4)\), the point is 3 units to the right of the origin and 4 units up.
Coordinate System Conversion
Converting between polar and rectangular coordinates involves using trigonometric relationships. Here's how you can switch between the two:
  • From rectangular to polar coordinates: Given \((x, y)\), to find \(r\) and \(\theta\): \[r = \sqrt{x^2 + y^2}\] \[\theta = \tan^{-1}(\frac{y}{x})\]

    • For instance, for the point \((3, 4)\), we get: \[r = \sqrt{3^2 + 4^2} = 5\] and \[\theta = \tan^{-1}(\frac{4}{3})\]
  • From polar to rectangular coordinates: Given \((r, \theta)\), to find \(x\) and \(y\): \[ x = r \cos(\theta) \] \[ y = r \sin(\theta) \]

    • For example, for the point \((5, \frac{\pi}{4})\), the conversion gives: \[ x = 5 \cos(\frac{\pi}{4}) = 3.54\] \[ y = 5 \sin(\frac{\pi}{4}) = 3.54\]

In this exercise, we converted the rectangular equation \(x = 4\) to its polar form. We utilized the relationship \( x = r \cos(\theta) \), replacing \( x \) with \( r \cos(\theta) \) to get: \[ r \cos(\theta) = 4 \]
This equation is the polar equivalent of the given rectangular equation.

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