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Chapter 10: Problem 1

Express the polar equation \(r=f(\theta)\) in parametric form in Cartesian coordinates, where \(\theta\) is the parameter.

Short Answer

Expert verified
Answer: The parametric equations in Cartesian coordinates are given by \(x = f(\theta)\cos(\theta)\) and \(y = f(\theta)\sin(\theta)\) with \(\theta\) as the parameter.

Step by step solution

01

Identify the polar equation

Given, the polar equation is \(r=f(\theta)\).
02

Convert from polar to Cartesian coordinates

To convert from polar to Cartesian coordinates, we use the following formulas: \(x = r\cos(\theta)\), \(y = r\sin(\theta)\). Since \(r=f(\theta)\), we can substitute the given function \(f(\theta)\) into these formulas to obtain the parametric equations in Cartesian coordinates.
03

Substitute the function f(胃) into the formulas

We replace \(r\) with \(f(\theta)\) in our conversion formulas: \(x = f(\theta)\cos(\theta)\), \(y = f(\theta)\sin(\theta)\). These are the parametric equations in Cartesian coordinates with \(\theta\) as the parameter.

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Key Concepts

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

Polar Coordinates
Polar coordinates offer a unique way to represent points in a plane using two values: the distance from the origin (known as the radius, or "r") and the angle ("\(\theta\)") formed with the positive x-axis. This system is particularly useful for dealing with circular or spiral patterns where Cartesian coordinates might be cumbersome.

In the polar coordinate system:
  • \(r\) represents the radial distance from the origin to the point.
  • \(\theta\) is the angle measured in a counter-clockwise direction from the positive x-axis.

Unlike the rectangular grid of Cartesian coordinates, the grid formed by polar coordinates is circular, making it particularly useful in fields involving rotations, such as engineering and physics.

Understanding polar coordinates allows us to switch perspectives between these systems, leveraging the strengths of each.
Cartesian Coordinates
Cartesian coordinates describe a point in a two-dimensional plane using an "x" and "y" value. These coordinates form the backbone of most geometry studied in school. They define a straightforward way to describe any location on a flat surface through a horizontal and vertical axis.

The key features are:
  • The x-coordinate measures horizontal distance from the origin.
  • The y-coordinate measures vertical distance from the origin.

The simplicity and familiarity of Cartesian coordinates make them ideal for linear calculations and designing graphs.

When converting between polar and Cartesian, these coordinates provide the rectangular lens through which we can reinterpret circular paths and relationships into straight lines.
Conversion Formulas
Converting polar coordinates to Cartesian coordinates involves simple trigonometric relationships that help bring curved paths into a linear perspective. This conversion is essential for analyzing complex shapes and motion using straightforward algebraic techniques.

Here are the key conversion formulas:
  • \(x = r\cos(\theta)\) translates the radius and angle into the x-axis coordinate.
  • \(y = r\sin(\theta)\) translates the radius and angle into the y-axis coordinate.
These conversions allow us to express any polar equation like \(r=f(\theta)\) in Cartesian terms by substituting the function \(f(\theta)\) for \(r\).

By understanding and applying these formulas, we can easily switch between polar and Cartesian systems, depending on which best suits our current problem. This skill helps in solving mathematical problems involving curves, circles, and complex shapes with ease.

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Most popular questions from this chapter

Find an equation of the following curves, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, asymptotes (if they exist), and directrices. Use a graphing utility to check your work. An ellipse with vertices (卤9,0) and eccentricity \(\frac{1}{3}\)

The Lam茅 curve described by \(\left|\frac{x}{a}\right|^{n}+\left|\frac{y}{b}\right|^{n}=1,\) where \(a, b,\) and \(n\) are positive real numbers, is a generalization of an ellipse. a. Express this equation in parametric form (four pairs of equations are needed). b. Graph the curve for \(a=4\) and \(b=2,\) for various values of \(n\) c. Describe how the curves change as \(n\) increases.

An ellipse (discussed in detail in Section 10.4 ) is generated by the parametric equations \(x=a \cos t, y=b \sin t.\) If \(0 < a < b,\) then the long axis (or major axis) lies on the \(y\) -axis and the short axis (or minor axis) lies on the \(x\) -axis. If \(0 < b < a,\) the axes are reversed. The lengths of the axes in the \(x\) - and \(y\) -directions are \(2 a\) and \(2 b,\) respectively. Sketch the graph of the following ellipses. Specify an interval in t over which the entire curve is generated. $$x=4 \cos t, y=9 \sin t$$

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. When \(t=0,\) Earth is at (2,0) and Mars is at (3,0) both orbit the Sun (at (0,0) ) in the counterclockwise direction. The position of Mars relative to Earth is given by the parametric equations \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\) a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars relative to Earth is a lima莽on (Exercise 89).

Consider the curve \(r=f(\theta)=\cos a^{\theta}-1.5\) where \(a=(1+12 \pi)^{1 /(2 \pi)} \approx 1.78933\) (see figure). a. Show that \(f(0)=f(2 \pi)\) and find the point on the curve that corresponds to \(\theta=0\) and \(\theta=2 \pi\) b. Is the same curve produced over the intervals \([-\pi, \pi]\) and \([0,2 \pi] ?\) c. Let \(f(\theta)=\cos a^{\theta}-b,\) where \(a=(1+2 k \pi)^{1 /(2 \pi)}, k\) is an integer, and \(b\) is a real number. Show that \(f(0)=f(2 \pi)\) and that the curve closes on itself. d. Plot the curve with various values of \(k\). How many fingers can you produce?
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