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Chapter 10: Problem 11

Express the given equations in polar coordinates. (a) \(x=3\) (b) \(x^{2}+y^{2}=7\) (c) \(x^{2}+y^{2}+6 y=0\) (d) \(9 x y=4\)

Short Answer

Expert verified
(a) \(r \cos \theta = 3\); (b) \(r = \sqrt{7}\); (c) \(r = -6 \sin \theta\); (d) \(9r^2 \cos \theta \sin \theta = 4\).

Step by step solution

01

Convert Equation (a) to Polar Coordinates

The equation given is \(x = 3\). In polar coordinates, \(x\) is converted using the relation \(x = r \cos \theta\). Thus, we have the equation in polar form: \(r \cos \theta = 3\).
02

Convert Equation (b) to Polar Coordinates

The equation given is \(x^2 + y^2 = 7\). In polar form, \(x^2 + y^2\) is equal to \(r^2\), so we have \(r^2 = 7\). Thus, the polar coordinate form is \(r = \sqrt{7}\).
03

Convert Equation (c) to Polar Coordinates with Complete Square

The given equation is \(x^2 + y^2 + 6y = 0\). First, rewrite \(y^2 + 6y\) as \((y+3)^2 - 9\). Thus, the equation becomes \(x^2 + (y+3)^2 - 9 = 0\). Rearrange to get \(x^2 + (y+3)^2 = 9\). In polar coordinates, this means \(r^2 = 9\) and expands to \(r \cos \theta = 0\) but after substituting back, we need \(x^2 + y^2 + 6r \sin \theta = 0\) leading to \(r = -6 \sin \theta\).
04

Convert Equation (d) to Polar Coordinates

The given equation is \(9xy = 4\). Replace \(x\) and \(y\) with their polar equivalents: \(x = r \cos \theta\) and \(y = r \sin \theta\). The equation becomes \(9(r \cos \theta)(r \sin \theta) = 4\) or \(9r^2 \cos \theta \sin \theta = 4\).

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Key Concepts

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

Equation Conversion
In mathematics, equation conversion is a key process that involves changing the format of equations to better understand or solve them. When dealing with polar coordinates, it's common to convert Cartesian equations (using x and y) into polar form (using r and \( \theta \)). Here鈥檚 how this conversion typically works:

  • For any point on the plane, you can express x and y as \( x = r \cos \theta \) and \( y = r \sin \theta \).
  • An equation like \( x = 3 \) converts to \( r \cos \theta = 3 \) in polar coordinates.
  • For circles and ellipse equations like \( x^2 + y^2 = 7 \), simply recognize this as \( r^2 = 7 \) in polar terms.
  • Completing the square is a technique useful in converting more complex equations like \( x^2 + y^2 + 6y = 0 \), which helps simplify the conversion process.
  • Multiplicative equations, such as \( 9xy = 4 \), involve substituting both x and y to equate in polar terms, often resulting in squared terms.
Understanding these basic conversions is crucial as it allows for the manipulation and simplification of equations into a form that can be graphically represented or analytically solved in a polar coordinate system.
Coordinate Systems
Coordinate systems are ways to identify and locate points within a particular space. In typical real-world problems, two primary types of coordinate systems are used: Cartesian coordinates and polar coordinates.

  • **Cartesian Coordinates**: These use two axes, x and y, at right angles to each other to determine the location of a point in a plane. Each point in a 2D space can be described using an ordered pair (x, y).
  • **Polar Coordinates**: These use a point鈥檚 distance from a reference pole (usually the origin) and an angle from a reference direction (often the positive x-axis). Points are expressed using (r, \( \theta \)), where r is the radial distance and \( \theta \) is the angular displacement.
  • Each system has its advantages: the Cartesian system is straightforward for linear equations, while polar coordinates simplify the graph and solve problems involving circles and angles.
  • Converting between these systems enables more effortless computation and problem-solving in various mathematical fields.
Recognizing when to use each system, and how to convert between them, enhances problem-solving efficiency and mathematical comprehension.
Trigonometric Functions
Trigonometric functions are foundational in mathematics, especially when dealing with angles and circles, often found in polar coordinates. Here鈥檚 a simple breakdown of how these functions relate to polar coordinates:

  • Common functions include sine (sin), cosine (cos), and tangent (tan), which relate the angles and sides of a triangle.
  • In the context of polar coordinates, \( x = r \cos \theta \) suggests using cosine to find the x-coordinate from the hypotenuse (r) and the angle (\( \theta \)).
  • Similarly, \( y = r \sin \theta \) employs the sine function to determine the y-coordinate.
  • The equation \( x^2 + y^2 = r^2 \) is a fundamental identity involving trigonometric functions, as it directly associates with these conversions.
  • Utilizing these trigonometric relationships simplifies the process of transforming equations and gives rise to unique solutions to otherwise cumbersome calculations.
Trigonometric functions are more than just tools for calculation; they are central to understanding and connecting various branches of mathematics.

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Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. If \(d\) is a positive constant, then the conic section with polar equation $$ r=\frac{d}{1+\cos \theta} $$ is a parabola.

Use a graphing utility to generate the graph of the bifolium \(r=2 \cos \theta \sin ^{2} \theta\), and find the area of the upper loop.

Show that the curve with parametric equations $$ x=t^{2}-3 t+5, \quad y=t^{3}+t^{2}-10 t+9 $$ intersects itself at the point \((3,1)\), and find equations for the two tangent lines to the curve at the point of intersection.

Find the area inside the curve \(r^{2}=\sin 2 \theta\).

Find the highest point on the cardioid \(r=1+\cos \theta\).
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