Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the polar equation for the curve described by the following Cartesian equations. (a) \(y=45\) (b) \(x^{2}+y^{2}=36\) (c) \(x^{2}-y^{2}=1\) (d) \(4 x y=1\) (e) \(y=3 x+2\) (f) \(3 x^{2}+4 y=2\) (g) \(x^{2}+2 x+y^{2}-4 y-25=0\) Computers and graphing calculators offer a wonderful opportunity to experiment with the graphing of polar equations of the form \(r=f(\theta) .\) In some cases these aids require that the equations be recast in a parametric form. Since \(x=r \cos \theta=f(\theta) \cos \theta\) and \(y=r \sin \theta=f(\theta) \sin \theta\), you can use the parametric graphing capabilities to graph \(x=f(t) \cos t\) and \(y=f(t) \sin t\) as a set of parametric equations.

Step-by-Step Solution

Answer

(a) \(r = \frac{45}{\sin \theta}\), (b) \(r = 6\), (c) \(r = \pm \frac{1}{\cos 2\theta}\), (d) \(r = \pm \frac{1}{\sqrt{2\sin 2\theta}}\), (e) \(r = \frac{2}{\sin \theta - 3 \cos \theta}\), (f) Polar form remains complex, (g) Transform to \((r \cos \theta + 1)^2 + (r \sin \theta - 2)^2 = 30\).

Step 1: Understand the Cartesian Equation (a)

The Cartesian equation given is simple: \(y = 45\). This is a horizontal line in the Cartesian plane.

Step 2: Convert to Polar Form for (a)

In polar coordinates, \(y = r \sin \theta\). So, given \(y = 45\), we can represent it as \(r \sin \theta = 45\). Therefore, the polar equation is \(r = \frac{45}{\sin \theta}\).

Step 3: Understand the Cartesian Equation (b)

The equation \(x^2 + y^2 = 36\) represents a circle centered at the origin with radius 6 in Cartesian coordinates.

Step 4: Convert to Polar Form for (b)

In polar coordinates, \(x^2 + y^2 = r^2\). Therefore, the given equation becomes \(r^2 = 36\), leading to \(r = 6\).

Step 5: Understand the Cartesian Equation (c)

The equation \(x^2 - y^2 = 1\) signifies a hyperbola in Cartesian coordinates.

Step 6: Convert to Polar Form for (c)

Insert \(x = r \cos \theta\) and \(y = r \sin \theta\) into the equation \(x^2 - y^2 = 1\), leading to \(r^2 \cos^2 \theta - r^2 \sin^2 \theta = 1\), or \(r^2 (\cos^2 \theta - \sin^2 \theta) = 1\). Therefore, \(r = \pm \frac{1}{\cos 2\theta}\).

Step 7: Understand the Cartesian Equation (d)

The equation \(4xy = 1\) involves both \(x\) and \(y\), representing a hyperbola in Cartesian coordinates.

Step 8: Convert to Polar Form for (d)

Replace \(x = r \cos \theta\) and \(y = r \sin \theta\) to get \(4r \cos \theta \cdot r \sin \theta = 1\), leading to \(r^2 \sin 2\theta = \frac{1}{2}\). Thus, \(r = \pm \frac{1}{\sqrt{2\sin 2\theta}}\).

Step 9: Understand the Cartesian Equation (e)

The line \(y = 3x + 2\) is a linear equation in slope-intercept form with a slope of 3 and a y-intercept of 2.

Step 10: Convert to Polar Form for (e)

Substitute \(y = r \sin \theta\) and \(x = r \cos \theta\) to get \(r \sin \theta = 3r \cos \theta + 2\). Therefore, \(r (\sin \theta - 3 \cos \theta) = 2\). Thus, \(r = \frac{2}{\sin \theta - 3 \cos \theta}\).

Step 11: Understand the Cartesian Equation (f)

For \(3x^2 + 4y = 2\), it’s a quadratic equation in \(x\) with a linear term in \(y\).

Step 12: Convert to Polar Form for (f)

Substitute \(x = r \cos \theta\) and \(y = r \sin \theta\) to get \(3r^2 \cos^2 \theta + 4r \sin \theta = 2\). So, \(r (3r \cos^2 \theta + 4 \sin \theta) = 2\). This equation remains complex as it is.

Step 13: Understand the Cartesian Equation (g)

The equation \(x^2 + 2x + y^2 - 4y - 25 = 0\) forms a conic. First, complete the square to identify its shape.

Step 14: Complete the Square for (g)

Rewrite the equation as \((x+1)^2 + (y-2)^2 = 30\). It now represents a circle centered at \((-1, 2)\) with a radius of \(\sqrt{30}\).

Step 15: Convert to Polar Form for (g)

Using \(x = r \cos \theta\) and \(y = r \sin \theta\), write \((r \cos \theta + 1)^2 + (r \sin \theta - 2)^2 = 30\). This transformation to polar coordinates results in the given equation.

Key Concepts

Cartesian equations

Cartesian equations are mathematical expressions that establish relationships between variables using the Cartesian coordinate system. In this system, a point in a plane is defined by two independent rectangular coordinates: x and y. Each variable corresponds to a specific distance along the x-axis and y-axis.
These equations can describe various types of geometric shapes:

• **Lines**: A linear equation like \(y = mx + b\) describes a straight line with slope \(m\) and y-intercept \(b\).
• **Circles**: An equation such as \(x^2 + y^2 = r^2\) identifies a circle centered at the origin with radius \(r\).
• **Hyperbolas**: Equations of the form \(x^2 - y^2 = c\) reflect hyperbolas, which are open curves.

Understanding Cartesian equations can help you visualize and solve problems concerning the spatial arrangement of geometric figures on a plane.

coordinate conversion

Transforming Cartesian equations into polar equations requires a method called coordinate conversion. This process entails changing the reference frame from rectangular coordinates (x, y) to polar coordinates (r, \(\theta\)).
Polar coordinates define a location in a plane using:

• **r**: the radial distance from the origin (0,0) to a point.
• **\(\theta\)**: the angle formed with the positive x-axis.

Conversion formulas are essential to this process:

• **x-coordinate**: \(x = r \cos \theta\)
• **y-coordinate**: \(y = r \sin \theta\)
• **For distance**: \(r = \sqrt{x^2 + y^2}\)
• **For angle**: \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

By using these formulas, you can effectively change the representation of a geometric figure from Cartesian to polar form, making it simpler to work with in certain scenarios, especially in trigonometry and calculus.

conic sections

Conic sections refer to the curves obtainable by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. These curves hold essential applications in fields ranging from architecture to celestial mechanics.
To understand them in Cartesian terms, consider:

• **Circle**: A set of points equidistant from a center point, described by \(x^2 + y^2 = r^2\).
• **Ellipse**: An extended circle by axes having different lengths, represented by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
• **Hyperbola**: A graph showing two symmetrical open curves, described by \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
• **Parabola**: A u-shaped curve, characterized by \(y^2 = 4ax\) or \(x^2 = 4ay\).

Understanding these shapes' Cartesian equations helps in transitioning them into polar or other coordinate systems, allowing for more comprehensive analyses and geometric modeling.

graphing calculators

Graphing calculators are powerful tools that allow for the visualization of complex mathematical equations. These devices can plot graphs, solve equations, and perform symbolic algebra, helping you understand functions in both Cartesian and polar forms.
When working with polar equations specifically, graphing calculators enable:

• **Immediate visualization**: When inputting polar equations, you can instantly see how variables affect the graph's shape.
• **Exploration of functions**: By adjusting parameters, you can explore a wide range of mathematical behaviors without manual calculations.
• **Parametric equations**: You can enter equations in parametric form, using calculations like \(x = f(t) \cos t\) and \(y = f(t) \sin t\) to explore polar graphs.

While this technology supports learning, it's crucial to understand the underlying mathematics to interpret the graphical displays correctly. This knowledge ensures you derive meaningful insights from the graphical output, enhancing your comprehension of coordinate systems and functional relationships.