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Chapter 9: Problem 27

Convert the polar equation to a rectangular equation. \(2 r \cos \theta+3 r \sin \theta=6\)

Short Answer

Expert verified
The rectangular equation corresponding to the given polar equation is \(2x + 3y = 6\).

Step by step solution

01

Write down the given polar equation

First, write down the given polar equation. \(2r\cos\theta + 3r\sin\theta = 6\)
02

Use the conversion formulas

Use the conversion formulas, \(x = r\cos\theta\) and \(y = r\sin\theta\), to replace the polar coordinates with rectangular coordinates: \(2r(\frac{x}{r}) + 3r(\frac{y}{r}) = 6\)
03

Simplify the equation

Simplify the equation by canceling out the common factors: \(2x + 3y = 6\) Now the equation is in rectangular form. So, the rectangular equation corresponding to the given polar equation is \(\boxed{2x + 3y = 6}\).

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Key Concepts

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

Polar Coordinates
Polar coordinates provide a unique way to locate points in a plane using a distance and an angle. Each point is identified with a pair \( r, \theta \). Here, \( r \) represents the radial distance from the origin (or pole), and \( \theta \) is the angle made with the positive x-axis.
  • Radial Distance (\( r \)): This is the direct distance between the point and the origin. It's always non-negative.
  • Angle (\( \theta \)): Measured in radians or degrees, this angle specifies the point's direction from the origin.
Polar coordinates are often used where circular or rotational symmetry simplifies the mathematical description. For example, plotting a location on a spiral path is more intuitive with polar coordinates.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are expressed as pairs \( x, y \) and are the most common means of expressing points on a plane. They represent how far a point is along horizontal (\( x \)) and vertical (\( y \)) axes.
  • \( x \)-coordinate: Reflects the horizontal positioning of the point. Positive values extend right of the origin, while negative values extend left.
  • \( y \)-coordinate: Reflects the vertical positioning. Positive values are above the origin, while negative values are below.
Rectangular coordinates provide a straightforward, grid-based representation, which makes them suitable for algebraic calculations and graphing in most computational systems.
Coordinate Transformation
The process of converting from polar to rectangular coordinates involves using specific formulas. This transformation is crucial when you need to switch between these two representations in problem-solving.
  • The formula for converting \( x \) is \( x = r \cos \theta \).
  • For \( y \), the formula is \( y = r \sin \theta \).
These transformations rely on basic trigonometry, where the radial distance projects onto the x and y axes through \( \cos \theta \) and \( \sin \theta \), respectively. Transformation allows us to leverage the simplicity of polar equations and convert them into a rectangular form for easier manipulation and interpretation.
Algebraic Manipulation
Algebraic manipulation is vital for converting and simplifying equations during coordinate transformations. It involves using basic algebraic rules and operations to reshape equations into a desired form.
  • Simplification: Reducing equations by canceling common factors or rearranging terms.
  • Substitution: Plugging specific values or expressions into a formula to obtain a solution.
In this exercise, after changing polar coordinates to rectangular, simplification converted \( 2r\cos\theta + 3r\sin\theta = 6 \) to \( 2x + 3y = 6 \). This illustrates how algebraic manipulation helps eliminate terms, making equations easier to solve or analyze.

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Most popular questions from this chapter

Find the area of the region that lies outside the first curve and inside the second curve. $$ r=1-\sin \theta, \quad r=1 $$

Find the area of the surface obtained by revolving the given curve about the given line. \(r=4 \cos \theta\) about the polar axis

a. Find an expression for the arc length of the curve defined by the parametric equations \(x=f^{\prime \prime}(t) \cos t+f^{\prime}(t) \sin t \quad y=-f^{\prime \prime}(t) \sin t+f^{\prime}(t) \cos t\) where \(a \leq t \leq b\) and \(f\) has continuous third-order derivatives. b. Use the result of part (a) to find the arc length of the curve \(x=6 t \cos t+3 t^{2} \sin t\) and \(y=-6 t \sin t+3 t^{2} \cos t\), where \(0 \leq t \leq 1\)

(a) plot the curve, and (b) find an approximation of its length accurate to two decimal places. \(r=0.2 \sqrt{\theta}+1\), where \(0 \leq \theta \leq 6 \pi \quad\) (parabolic spiral)

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The curve with parametric equations \(x=f(t)\) and \(y=g(t)\) is a line if and only if \(f\) and \(g\) are both linear functions of \(t\).
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