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Chapter 9: Problem 25

Convert the polar equation to a rectangular equation. \(r \cos \theta=2\)

Short Answer

Expert verified
The rectangular equation that corresponds to the given polar equation \(r \cos \theta = 2\) is \(x = 2\).

Step by step solution

01

Write down the given polar equation

We are given the polar equation: \( r\cos \theta = 2 \)
02

Use the relationship between polar and rectangular coordinates

Recall the relationship between polar and rectangular coordinates: \( x = r \cos \theta\) and \( y = r \sin \theta \)
03

Substitute the expression for x into the polar equation

Substitute the expression for x in terms of r and theta into the polar equation: \( r \cos \theta = 2 \) becomes \(x = 2\)
04

Write the final rectangular equation

The rectangular equation is simply: \(x = 2\) This is the rectangular equation that corresponds to the given polar equation, \(r \cos \theta = 2\).

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Key Concepts

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

Polar Coordinates
Polar coordinates are a way of representing points on a plane using angles and distances. Unlike the typical Cartesian or rectangular coordinates that use an
  • "x-coordinate" (horizontal axis)
  • "y-coordinate" (vertical axis)
polar coordinates are determined by two distinct components. These are:
  • Radius or radial distance ("r"): This is the distance of a point from the origin, also known as the pole.
  • Angle (theta ): Measured from the positive x-axis, this angle is often given in radians.
For example, in the polar coordinate (3, \frac{\pi}{4}):
  • The radius "r" is 3
  • and the angle \theta is \frac{\pi}{4}
This means the point is located 3 units away from the origin, along the path defined by the angle \( \frac{\pi}{4} \) in radians. Understanding this system is crucial for converting between different coordinate systems.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are used to specify points in a two-dimensional plane with two numerical values. These values correspond to distances from two perpendicular axes: the x-axis and the y-axis. In contrast to polar coordinates, rectangular coordinates require:
  • x-coordinate: The horizontal distance from the y-axis.
  • y-coordinate: The vertical distance from the x-axis.
These coordinates take the form ( x, y ) .
For instance, the coordinate ( 4, 5 ) indicates a location where 4 units are moved horizontally (along the x-axis), and 5 units are moved vertically (along the y-axis).
Rectangular coordinates are most often used in algebra, calculus, and a variety of real-world applications, like graph plotting.
Coordinate Transformation
Coordinate transformation is the process of converting one set of coordinates into another. It is a fundamental process in mathematics and is especially useful when matching different ways of representing data. For polar and rectangular coordinates conversions:
  • From Polar to Rectangular:
    Use the formulas:
    • \( x = r \cos \theta \)
    • \( y = r \sin \theta \)
    These transformations allow you to find rectangular coordinates, "x" and "y", from polar coordinates, "r" and "\theta".For example, if you have a polar coordinate (5, \frac{\pi}{3}), substitute into the formulas to find:
    • \(x = 5 \cos(\frac{\pi}{3}) = 2.5\)
    • \(y = 5 \sin(\frac{\pi}{3}) = 4.33\)
  • From Rectangular to Polar:
    Use the reverse formulas:
    • \( r = \sqrt{x^2 + y^2} \)
    • \( \theta = \tan^{-1}(\frac{y}{x}) \)
    These calculations convert rectangular coordinates back into polar form.
Coordinate transformation provides flexibility in solving mathematical problems across different types of coordinate systems, making it a key skill in analytical geometry and applied mathematics.

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Most popular questions from this chapter

Find the length of the curve defined by the parametric equations. $$ x=2 t^{3 / 2}, \quad y=3 t+1 ; \quad 0 \leq t \leq 4 $$

a. Let \(f\) be a function with a continuous derivative in an interval \([\alpha, \beta]\). If the graph \(C\) of \(r=f(\theta)\) is traced exactly once as \(\theta\) increases from \(\alpha\) to \(\beta\), show that the rectangular coordinates of the centroid of \(C\) are $$ \bar{x}=\frac{\int_{\alpha}^{\beta} r \cos \theta \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta}{\int_{\alpha}^{\beta} \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta} $$ and $$ \bar{y}=\frac{\int_{\alpha}^{\beta} r \sin \theta \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta}{\int_{\alpha}^{\beta} \sqrt{\left(r^{\prime}\right)^{2}+r^{2}} d \theta} $$ Hint: See the directions for Exercises 47 and 48 in Section \(5.7\). b. Use the result of part (a) to find the centroid of the upper semicircle \(r=a\), where \(a>0\) and \(0 \leq \theta \leq \pi\).

Use Equation (5) or Equation (6) to find the eccentricity of the conic with the given rectangular equation. \(9 x^{2}+25 y^{2}=225\)

Find all points of intersection of the given curves. \(r=\cos \theta \quad\) and \(\quad r=\cos 2 \theta\)

Use Equation (5) or Equation (6) to find the eccentricity of the conic with the given rectangular equation. \(\frac{x^{2}}{5}-\frac{y^{2}}{3}=1\)
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