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Chapter 11: Problem 31

Convert each polar equation to a rectangular equation. \(r=\frac{8}{2-\sin \theta}\)

Short Answer

Expert verified
\(2 \sqrt{x^2 + y^2} = y + 8\)

Step by step solution

01

Use the relationship between polar and rectangular coordinates

Recall the relationship between polar and rectangular coordinates: \(x = r \, \text{cos} \, \theta\) \(y = r \, \text{sin} \, \theta\) \(r^2 = x^2 + y^2\)
02

Substitute known values

The given polar equation is \(r = \frac{8}{2 - \text{sin} \, \theta}\). Multiply both sides by \(2 - \text{sin} \, \theta\) to clear the fraction: \(r(2 - \text{sin} \, \theta) = 8\)
03

Express sin(胃) in terms of rectangular coordinates

Using the relationship \(y = r \, \text{sin} \, \theta\), we get \(\text{sin} \, \theta = \frac{y}{r}\). Substitute this into the equation from Step 2: \(r \left(2 - \frac{y}{r}\right) = 8\)
04

Simplify the equation

Distribute \(r\) on the left-hand side: \(2r - y = 8\)
05

Solve for r

Recall from Step 1 that \(r = \sqrt{x^2 + y^2}\). Substitute \(r = \sqrt{x^2 + y^2}\) into the simplified equation: \(2 \sqrt{x^2 + y^2} - y = 8\)
06

Arrange into the standard form

Isolate the term with the square root to one side: \(2 \sqrt{x^2 + y^2} = y + 8\). Finally, simplify the equation into rectangular form.

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Key Concepts

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

rectangular coordinates
Rectangular coordinates are a way to represent points in a plane using two distances from two perpendicular lines known as the x-axis and y-axis. Each point is defined by an ordered pair \(x, y\). The x-coordinate tells you how far to move left or right, and the y-coordinate tells you how far to move up or down. Rectangular coordinates are commonly used because they allow straightforward calculation using basic algebraic operations. You can visualize rectangular coordinates like a city grid, with streets running horizontally and avenues running vertically.
polar coordinates
Polar coordinates represent points in a plane using a distance from a fixed point (the origin) and an angle from a fixed direction (usually the positive x-axis). Each point is defined by \(r, \theta\), where \(r\) is the radial distance and \(\theta\)\ is the angle measured in radians or degrees.
Polar coordinates are particularly useful in scenarios involving circular or spiral patterns, as they simplify many mathematical descriptions and calculations. While rectangular coordinates frame a point by its horizontal and vertical distances, polar coordinates frame a point by how far it is from the origin and in what direction.
trigonometric identities
Trigonometric identities are equations that relate the trigonometric functions to one another. They are foundational to converting between polar and rectangular forms, as they allow for the expression of sines and cosines in terms of x and y coordinates.
The primary trigonometric identities used for polar to rectangular conversion are:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \tan^{-1} (y/x)\)
These identities help transform the polar equation into a form that only involves x and y, making it easier to work with in algebraic or geometric problems.
algebraic manipulation
Algebraic manipulation involves rearranging and simplifying equations using algebraic operations. After converting a polar equation to involve sines and cosines, you鈥檒l usually need to use algebra to further simplify or rearrange the terms.
In transforming polar to rectangular equations, common algebraic manipulations include:
  • Multiplying both sides of an equation to eliminate fractions
  • Substituting trigonometric expressions with their rectangular equivalents
  • Isolating terms involving \(x\) and \(y\) on one side of the equation
For example, in the given problem, distributing \(r\) and then substituting \(r = \sqrt{x^2 + y^2}\) simplifies the equation into a rectangular form.
coordinate transformation
Coordinate transformation is the process of converting coordinates from one system to another, such as from polar to rectangular coordinates. This transformation is essential in contexts where one coordinate system may simplify the physics or geometry of a problem.
To convert from polar to rectangular coordinates, you use the established relationships between \(x, y, r,\) and \(\theta\). In our example, the steps follow a logical order:
  • Express the given polar equation as a function of \(r\) and \(\theta\)
  • Replace sines and cosines with their rectangular form equivalents
  • Resolve the resulting equation to express \(r\) using \(x, y\)
  • Finally, solve any remaining algebraic expressions to purely involve x and y
Such transformations allow solving complex problems by leveraging the strengths of different coordinate systems.

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Most popular questions from this chapter

Bill's train leaves at 8: 06 an and accelerates at the rate of 2 meters per second per second. Bill, who can run 5 meters per second, arrives at the train station 5 seconds after the train has left and runs for the train. (a) Find parametric equations that model the motions of the train and Bill as a function of time. [Hint: The position \(s\) at time \(t\) of an object having acceleration \(a\) is \(s=\frac{1}{2} a t^{2}\). (b) Determine algebraically whether Bill will catch the train. If so, when? (c) Simulate the motion of the train and Bill by simultaneously graphing the equations found in part (a).

An arch for a bridge over a highway is in the form of half an ellipse. The top of the arch is 20 feet above the ground level (the major axis). The highway has four lanes, each 12 feet wide; a center safety strip 8 feet wide; and two side strips, each 4 feet wide. What should the span of the arch be (the length of its major axis) if the height 28 feet from the center is to be 13 feet?

Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas \(\frac{x^{2}}{4}-y^{2}=1 \quad\) and \(\quad y^{2}-\frac{x^{2}}{4}=1\) are conjugate. Graph each hyperbola on the same set of coordinate axes.

A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center?

The left field wall at Fenway Park is 310 feet from home plate; the wall itself (affectionately named the Green Monster) is 37 feet high. A batted ball must clear the wall to be a home run. Suppose a ball leaves the bat 3 feet above the ground, at an angle of \(45^{\circ} .\) Use \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\) as the acceleration due to gravity, and ignore any air resistance. (a) Find parametric equations that model the position of the ball as a function of time. (b) What is the maximum height of the ball if it leaves the bat with a speed of 90 miles per hour? Give your answer in feet. (c) How far is the ball from home plate at its maximum height? Give your answer in feet. (d) If the ball is hit straight down the left field line, will it clear the Green Monster? If it does, by how much does it clear the wall?
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