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Chapter 8: Problem 55

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. $$ r^{2}=9 \cos (2 \theta) $$

Short Answer

Expert verified
The equation represents a lima莽on in polar form, and a complex quartic in rectangular form.

Step by step solution

01

Recall Polar to Rectangular Formulas

To convert from polar to rectangular equations, we use the relationships: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). In this exercise, we start with the equation \( r^2 = 9 \cos(2\theta) \).
02

Use Trigonometric Identity

Recall the double angle identity for cosine: \( \cos(2\theta) = 2\cos^2\theta - 1 \). Substituting this into our equation gives us \( r^2 = 9(2\cos^2\theta - 1) \).
03

Substitute Rectangular Coordinates

Substitute the polar to rectangular conversions: \( r^2 = x^2 + y^2 \) and \( \cos^2\theta = \left(\frac{x}{r}\right)^2 = \frac{x^2}{r^2} \). This simplifies the equation to \( x^2 + y^2 = 18\left(\frac{x^2}{x^2 + y^2}\right) - 9 \).
04

Clear Denominators and Rearrange

Multiply both sides by \( x^2 + y^2 \) to eliminate the fraction: \( (x^2 + y^2)^2 = 18x^2 - 9(x^2 + y^2) \). Simplify to get \( x^4 + 2x^2y^2 + y^4 = 9x^2 - 9y^2 \).
05

Simplify Further

By rearranging terms: \( x^4 + y^4 + 2x^2y^2 + 9y^2 - 9x^2 = 0 \). This equation doesn't simplify readily to a common conic, indicating it represents a lima莽on when considered in polar form, or a complex quartic in rectangular form.

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Key Concepts

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

Double Angle Identity
The double angle identity is a powerful trigonometric tool that simplifies expressions involving trigonometric functions of double angles. Specifically, for the cosine of double angles, the double angle identity is given by:\[\cos(2\theta) = 2\cos^2(\theta) - 1\]This identity is crucial when converting equations from polar to rectangular form, as it helps in expressing functions of \(\theta\) using basic trigonometric functions.
  • It is particularly useful in simplifying and reducing complexities in trigonometric expressions.
  • Allows substitution of double angle terms with simpler expressions that are easier to manipulate algebraically.
By applying the double angle identity to the given equation \(r^{2}=9\cos(2\theta)\), we substitute \(\cos(2\theta)\) with \(2\cos^2(\theta) - 1\), making the equation more manageable for further conversion into rectangular coordinates.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a 2D plane using a pair of values (\(x, y\)). In the context of this exercise, we need to convert polar coordinates, given in terms of \(r\) and \(\theta\), into rectangular coordinates. This conversion is facilitated by certain relationships:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
  • \(r^2 = x^2 + y^2\)
These formulas help convert an equation from polar form, where positions are specified by a radius and an angle, into rectangular form, where positions are specified by horizontal and vertical positions.In our exercise, after substituting \(\cos^2\theta\) with \(\left(\frac{x}{r}\right)^2\), we manipulate the equation using these identities, ultimately converting it into a quartic equation, \(x^4 + y^4 + 2x^2y^2 + 9y^2 - 9x^2 = 0\). This demonstrates the versatility in using different coordinate systems to represent complex equations.
Conic Sections
Conic sections are the curves obtained by intersecting a plane with a cone. The basic types of conic sections include circles, ellipses, parabolas, and hyperbolas. They are commonly expressed using quadratic equations in rectangular coordinates. The interesting part about conic sections is their variations when expressed in polar coordinates. However, sometimes polar equations may represent more complex curves, like the lima莽on in this exercise. Converting to rectangular coordinates helps reveal the nature of the curve. In our solution, we attempted to categorize the resulting equation from polar form. While it doesn't simplify to a familiar conic section like a circle or parabola, by recognizing it as a lima莽on in polar form, we observe its unique characteristics.
  • This illustrates the beauty and complexity of conic sections beyond their typical forms.
  • Conic sections help in understanding the geometric significance of algebraic equations.
Overall, exploring the characteristics of conic sections allows students to see the link between simple geometric figures and more advanced mathematical concepts.

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Most popular questions from this chapter

For Exercises 73 and 74, refer to the following: The lemniscate motion occurs naturally in the flapping of birds' wings. The bird's vertical lift and wing sweep create the distinctive figure-eight pattern. The patterns vary with different wing profiles. Flapping Wings of Birds. Compare the two following possible lemniscate patterns by graphing them on the same polar graph: \(r^{2}=4 \cos (2 \theta)\) and \(r^{2}=4 \cos (2 \theta+2)\).

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. How far does the projectile travel (what is the horizontal distance), and what is its maximum altitude? (Note the symmetry of the projectile path.)

Actual Speed and True Course. An airplane is flying on a course of \(285^{\circ}\) as measured from due north at \(300 \mathrm{mph}\). The wind is blowing due south at \(30 \mathrm{mph}\). Represent their respective vectors as complex numbers written in polar form, and determine the resultant speed and direction vector.

For Exercises 71 and 72, refer to the following: Spirals are seen in nature-for example, in the swirl of a pine cone. They are also used in machinery to convert motions. An Archimedes spiral has the general equation \(r=a \theta\). A more general form for the equation of a spiral is \(r=a \theta^{1 / n}\), where \(n\) is a constant that determines how tightly the spiral is wrapped. Archimedes Spiral. Compare the Archimedes spiral \(r=\theta\) with the spiral \(r=\theta^{1 / 2}\) by graphing both on the same polar graph.

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. Suppose the ride conductor was rather sinister and speeded up the ride to twice the speed. How would you modify the parametric equations to model such a change? Now vary the values of \(A\) and \(B\). What do you conjecture these parameters are modeling in this problem?
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