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Chapter 11: Problem 87

Convert the equation from polar coordinates into rectangular coordinates. \(r=-2 \sin (\theta)\)

Short Answer

Expert verified
The rectangular form is \(x^2 + y^2 + 2y = 0\).

Step by step solution

01

Identify the polar equation

The given equation is in polar form: \(r = -2 \sin(\theta)\). This is in terms of \(r\) and \(\theta\).
02

Recall the polar to rectangular conversion formulas

To convert from polar to rectangular coordinates, use the relationships: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). Also, remember: \(r^2 = x^2 + y^2\) and \(\sin(\theta) = \frac{y}{r}\).
03

Replace \(\sin\) using its conversion relation

Since \(\sin(\theta) = \frac{y}{r}\), we can substitute \(-2 \sin(\theta)\) as: \(r = -2 \frac{y}{r}\).
04

Simplify by eliminating \(r\)

Multiply both sides by \(r\) to get rid of the fraction: \(r^2 = -2y\).
05

Substitute \(r^2\) using \(x\) and \(y\)

Replace \(r^2\) with \(x^2 + y^2\): \(x^2 + y^2 = -2y\).
06

Rearrange into standard form

Move all terms to one side for standard form: \(x^2 + y^2 + 2y = 0\).
07

Confirm the solution

The equation is now in rectangular coordinates. Ensure it reflects a standard form. This can represent a circle, shifted downward, although it does not form a standard circle equation with a clear center and radius.

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Key Concepts

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

Polar Coordinates
Polar coordinates offer a way to describe a point's location based on its distance from the origin and the angle it forms with the positive x-axis. We use two primary values:
  • \( r \): The radial coordinate, which is the distance from the origin.
  • \( \theta \): The angular coordinate, which is the angle from the positive x-axis.
This coordinate system is particularly useful in scenarios involving circular or rotational motion, where angles and distances from a fixed point are more relevant than rectangular axes. To visualize, picture plotting points around a circle, each defined by how far you go from the center (which is the radius, \( r \)) and in what direction \( (\theta) \) you are facing.
Rectangular Coordinates
Contrary to polar coordinates, rectangular coordinates rely on horizontal and vertical axes to describe the location of points in a plane. Here, coordinates are expressed as:
  • \( x \): A point's horizontal position, either left or right of the origin.
  • \( y \): A point's vertical position, above or below the origin.
Think of rectangular coordinates as the kind you typically encounter in algebra courses, where each point is plotted as \((x, y)\) on a grid. This method makes it easy to calculate distances and angles using basic algebraic principles.
Coordinate Conversion
Converting between polar and rectangular coordinates involves using some simple trigonometric relationships. To perform this conversion:
  • From polar to rectangular: Use the equations \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).
  • From rectangular to polar: Use \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan(\frac{y}{x})\).
When transferring the equation \(r = -2\sin(\theta)\) from polar to rectangular form, we use these relationships to resolve the components along the x and y axes. The conversion ensures that the equation can be interpreted and graphed in a rectangular coordinate frame.
Trigonometric Identities
Trigonometric identities play a crucial role in coordinate conversion. They allow us to express trigonometric functions like sine and cosine in terms of x and y. Key identities include:
  • \(\sin(\theta) = \frac{y}{r}\)
  • \(\cos(\theta) = \frac{x}{r}\)
These identities facilitate the manipulation of equations, such as transforming the polar equation \(r = -2 \sin(\theta)\) into a more familiar rectangular equation format. By substituting \(\sin(\theta)\) with \(\frac{y}{r}\), and then clearing fractions and simplifying, we align the equation with a standard rectangular representation.

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Most popular questions from this chapter

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