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Chapter 10: Problem 32

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph In an \(r \theta\) -plane. $$r=4 \sec \theta$$

Short Answer

Expert verified
The equation is \(x = 4\); it graphically represents a vertical line at \(x = 4\).

Step by step solution

01

Convert Secant to Cartesian Coordinates

The polar equation given is \( r = 4 \sec \theta \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \), so this is equivalent to \( r = \frac{4}{\cos \theta} \). Multiply through by \( \cos \theta \) to get \( r \cos \theta = 4 \).
02

Identify Cartesian Coordinates Relationships

We know the relationships between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). From the equation \( r \cos \theta = 4 \), substitute \( x \) to get \( x = 4 \).
03

Equation in Cartesian Coordinates

The final step is simply recognizing that the equation \( x = 4 \) is already in Cartesian coordinates. This describes a vertical line in the Cartesian coordinate plane at \( x = 4 \).
04

Graph the Equation

To sketch the graph in the \( r \theta \)-plane, consider that for \( x = 4 \), the radius remains constant for given \( \theta \), so \( r \) varies as \( \theta \) changes, resulting in a straight vertical line when seen from any point in space.

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Key Concepts

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

Polar Coordinates
Polar Coordinates are a system of describing points in a plane using a distance from a reference point and an angle from a reference direction. The reference point is known as the pole, and the angle is measured from the positive x-axis, wrapping counterclockwise.
In Polar Coordinates, we often use the variables \( r \) and \( \theta \).
  • \( r \) represents the radial distance from the pole.
  • \( \theta \) denotes the angle in radians.
When translating a point described in polar coordinates to the more familiar Cartesian Coordinates, you rely on the relationships:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
These equations help us convert polar equations to Cartesian equations, making it easier to visualize the graph on the standard x-y plane.
Secant Function
The Secant Function, represented as \( \sec \theta \), is closely related to the cosine function. Specifically, it is the reciprocal of the cosine function:
  • \( \sec \theta = \frac{1}{\cos \theta} \)
Understanding this relationship is key in converting certain polar equations into Cartesian form. For example, in the equation \( r = 4 \sec \theta \), substituting the secant function provides a different perspective:
  • The equation becomes \( r = \frac{4}{\cos \theta} \).
This conversion simplifies to \( r \cos \theta = 4 \), which is crucial for transitioning to Cartesian Coordinates. By understanding these trigonometric functions, we can navigate between different forms of equations.
Cartesian Coordinates
Cartesian Coordinates identify points in a plane using ordered pairs \((x, y)\). This method is based on two perpendicular axes intersecting at a point called the origin.
  • \( x \) represents the horizontal distance from the origin.
  • \( y \) indicates the vertical distance from the origin.
The translation of equations from polar to Cartesian coordinates involves using the relationships:
  • \( x = r \cos \theta \)
  • \( y = r \sin \theta \)
In the example provided, the polar equation \( r \cos \theta = 4 \) translates directly to \( x = 4 \), illustrating a vertical line in the Cartesian plane where every point has an x-coordinate of 4. The ability to recognize and convert these equations is essential for accurate graphing and interpretation.
Graphing Equations
Graphing Equations serves as a powerful visual tool in understanding mathematical relationships. Moving between polar and Cartesian systems can significantly change how we perceive a graph.
When graphing the equation \( x = 4 \) in Cartesian coordinates:
  • It appears as a vertical line crossing the x-axis at the point (4, y).
In the r-\( \theta \)-plane, this visualization adjusts because each point on the graph is represented in relation to the pole:
  • For various \( \theta \) values, the radius \( r \) changes to maintain the point on the line, perceived as \( x = 4 \).
Understanding graphing in both systems helps reinforce how different forms can represent the same mathematical relationship. Practice sketching graphs helps build stronger recognition and synthesis of equations in various forms.

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Most popular questions from this chapter

Graph the parabolas on the same coordinate plane, and estimate the points of intersection. $$y=x^{2}-2.1 x-1 ; \quad x=y^{2}+1$$

Graph the Lissajous figure in the viewing rectangle \([-1,1]\) by \([-1,1]\) for the specified range of \(t\). $$x(t)=\sin (4 t), \quad y(t)=\sin (3 t+\pi / 6) ; \quad 0 \leq t \leq 6.5$$

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure. $$\begin{array}{lll}C_{1:} x=1+\cos t, & y=1+\sin t ; & \pi / 3 \leq t \leq 2 \pi \\\C_{2}: x=1+\tan t, & y=1 ; & 0 \leq t \leq \pi / 4\end{array}$$

Find an equation for the set of points in an xy-plane that are equidistant from the point \(P\) and the line \(l\). $$P(5,-2) ; \quad l: y=4$$

Sketch the graph of the polar equation. $$r=1-\csc \theta$$
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