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Chapter 8: Problem 67

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2} \sin ^{2} \theta+2 r \cos \theta=3 $$

Short Answer

Expert verified
The polar equation converts to the rectangular equation \( y^2 + 2x = 3 \), which is a parabola.

Step by step solution

01

Understand Polar to Rectangular Conversion

To convert a polar equation to a rectangular form, recall the relations: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). Use these relationships to rewrite the given polar equation.
02

Isolate Trigonometric Terms

The polar equation is \( r^2 \sin^2 \theta + 2r \cos \theta = 3 \). Rewrite \( r \sin \theta \) as \( y \) and \( r \cos \theta \) as \( x \). This results in \( r^2 \sin^2 \theta = y^2 \) and \( 2r \cos \theta = 2x \). Substitute these into the equation.
03

Substitute and Simplify

Substitute into the equation: \( y^2 + 2x = 3 \). This is the rectangular form of the equation.
04

Identify the Resulting Equation Type

The equation \( y^2 + 2x = 3 \) resembles the standard parabola form \( y^{2} = 4ax \) after rearranging to \( y^2 = -2x + 3 \), which confirms that it is a parabola.

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Key Concepts

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

Trigonometric Identities
Trigonometric identities are essential tools when working with coordinate conversions, such as changing from polar to rectangular forms. In our task, we encounter the common identities:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
  • \( r^2 = x^2 + y^2 \)

These identities help us bridge polar coordinates, which are expressed in terms of \( r \) (radius) and \( \theta \) (angle), with rectangular coordinates \(x\) and \(y\).
Applying these identities means we're translating an equation that uses angles (\( \theta \)) and distances (\( r \)) into one that uses simple horizontal (\( x \)) and vertical (\( y \)) positions. Using this conversion unlocks the ability to analyze and graph equations more intuitively in a two-dimensional plane.
Rectangular Equations
Rectangular equations express relationships using \( x \) and \( y \) coordinates, which makes them more straightforward to handle and plot on the Cartesian plane. In our exercise, after conversion from the polar form, the equation \( y^2 + 2x = 3 \) is obtained.
This transformation occurs as follows:
  • Replace \( r \cos \theta \) with \( x \), making \( 2r \cos \theta = 2x \).
  • Convert \( r^2 \sin^2 \theta \) to \( y^2 \) since it is equivalent to \((r \sin \theta)^2 = y^2\).

After substitution, the polar equation turns into an easily understandable rectangular equation. Grasping how these transformations work helps demystify the process of dealing with polar coordinates and making them applicable in Cartesian graphs, which are commonly used in fields like physics and engineering.
Graphing Parabolas
Graphing parabolas on the Cartesian plane is a critical skill that allows for visual representation and analysis of quadratic relationships. In the resulting rectangular equation \( y^2 + 2x = 3 \), this identifies as a parabola.
To graph, you can rearrange it into the standard parabola form: \( y^2 = 4ax \). Rearranging this equation gives \( y^2 = -2x + 3 \) or equivalently \( y^2 = 4 \left( -\frac{1}{2} \right)x \).
This indicates a parabola that opens horizontally instead of the more common vertical orientation. Key features of this graph include:
  • The vertex is shifted along the x-axis.
  • It opens to the left because of the negative coefficient.

Understanding how to manipulate and recognize the form of parabola equations enables a student to efficiently sketch the graph and predict the path of the curve, a helpful visualization tool in various applications including physics and design.

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Most popular questions from this chapter

For Exercises 47 and 48 , refer to the following: Modern amusement park rides are often designed to push the envelope in terms of speed, angle, and ultimately \(g\)-force, and usually take the form of gargantuan roller coasters or skyscraping towers. However, even just a couple of decades ago, such creations were depicted only in fantasy-type drawings, with their creators never truly believing their construction would become a reality. Nevertheless, thrill rides still capable of nauseating any would-be rider were still able to be constructed; one example is the Calypso. This ride is a not-too-distant cousin of the better-known Scrambler. It consists of four rotating arms (instead of three like the Scrambler), and on each of these arms, four cars (equally spaced around the circumference of a circular frame) are attached. Once in motion, the main piston to which the four arms are connected rotates clockwise, while each of the four arms themselves rotates counterclockwise. The combined motion appears as a blur to any onlooker from the crowd, but the motion of a single rider is much less chaotic. In fact, a single rider's path can be modeled by the following graph: The equation of this graph is defined parametrically by $$ \begin{aligned} &x(t)=A \cos t+B \cos (-3 t) \\ &y(t)=A \sin t+B \sin (-3 t), 0 \leq t \leq 2 \pi \end{aligned} $$ Amusement Rides. What is the location of the rider at \(t=0, t=\frac{\pi}{2}, t=\pi, t=\frac{3 \pi}{2}\), and \(t=2 \pi\) ?

In Exercises 79 and 80, explain the mistake that is made. Convert the rectangular coordinate \((-\sqrt{3}, 1)\) to polar coordinates. Solution: Label \(x\) and \(y . \quad x=-\sqrt{3}, y=1\) Find \(r . \quad r=\sqrt{x^{2}+y^{2}}=\sqrt{3+1}=\sqrt{4}=2\) Find \(\theta\). $$ \begin{aligned} \tan \theta &=\frac{1}{-\sqrt{3}}=-\frac{1}{\sqrt{3}} \\ \theta &=\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)=-\frac{\pi}{4} \end{aligned} $$ Write the point in polar \(\quad\left(2,-\frac{\pi}{4}\right)\) coordinates. This is incorrect. What mistake was made?

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=\sqrt{t}, y=t+2 $$

In Exercises 1-20, graph the curve defined by the following sets of parametric equations. Be sure to indicate the direction of movement along the curve. $$ x=2 \sin (3 t), y=3 \cos (2 t), t \text { in }[0,2 \pi] $$

For Exercises 37-46, recall that the flight of a projectile can be modeled with the parametric equations $$ x=\left(v_{0} \cos \theta\right) t \quad y=-16 t^{2}+\left(v_{0} \sin \theta\right) t+h $$ where \(t\) is in seconds, \(v_{0}\) is the initial velocity in feet per second, \(\theta\) is the initial angle with the horizontal, and \(h\) is the initial height above ground, where \(x\) and \(y\) are in feet. Flight of a Projectile. A projectile is launched from the ground at a speed of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. After how many seconds does the projectile hit the ground?
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