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Chapter 12: Problem 64

Write the polar equation \(3 r=\sin \theta\) as an equation in rectangular coordinates. Identify the equation and graph it.

Short Answer

Expert verified
\[ 3x^2 + 3y^2 - y = 0 \] is a circle.

Step by step solution

01

Identify Polar Equation

Given polar equation: \[ 3r = \sin \theta \]
02

Express Variables

Express the variables in the polar equation in terms of rectangular coordinates. Recall that: \[ x = r \cos \theta \] \[ y = r \sin \theta \] and \[ r = \sqrt{x^2 + y^2} \]
03

Express \sin \theta

Since \sin \theta = \frac{y}{r}, rewrite the given equation \[ 3r = \sin \theta \] as \[ 3r = \frac{y}{r} \]
04

Isolate r

Isolate r by multiplying both sides by r: \[ 3r^2 = y \]
05

Substitute for r

Substitute \sqrt{x^2 + y^2} for r: \[ 3(x^2 + y^2) = y \]
06

Distribute and Rearrange

Distribute 3 and rearrange the equation to standard form: \[ 3x^2 + 3y^2 - y = 0 \]
07

Identify the Equation

Identify the resulting equation. This equation represents a conic section, specifically a circle.
08

Graph the Equation

Plot the graph of the equation \[ 3x^2 + 3y^2 - y = 0 \] This is a circle centered at the origin, but shifted slightly upwards.

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Key Concepts

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

Polar Coordinates
In polar coordinates, each point in a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (origin) is often denoted as \(O\), and the reference direction is usually the positive x-axis. The coordinates are given as \( (r, \theta) \), where:
  • \(r\): the radial distance from the origin to the point
  • \(\theta\): the angle measured counterclockwise from the positive x-axis to the line segment connecting the origin to the point
For example, in the given exercise, the polar equation is \(3r = \sin \theta\). By understanding polar coordinates, we focus on the distance and angle measurements.
Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes: the x-axis and the y-axis. Any point in the plane is represented as \( (x, y) \), where:
  • \(x\): the horizontal distance from the origin
  • \(y\): the vertical distance from the origin
Connecting these to polar coordinates, we have useful conversion formulas:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
  • \(r = \sqrt{x^2 + y^2}\)
Understanding rectangular coordinates helps in converting between different coordinate systems, as seen when turning \(3r = \sin \theta\) into \(3(x^2 + y^2) = y\).
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the angle involved. They are crucial in converting between coordinate systems. In particular:
  • \(\sin \theta = \frac{y}{r}\)
On transforming the exercise equation, we use \(\sin \theta = \frac{y}{r}\) to rewrite the polar equation \(3r = \sin \theta\) as \(3r = \frac{y}{r}\), simplifying it into a form that involves \(r\), which is then replaced by \(\r = \sqrt{x^2 + y^2}\).
Conic Sections
Conic sections are curves obtained by slicing a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas. The equation \(3(x^2 + y^2) = y\) is a form of a conic section. Specifically, simplifying and analyzing it reveals that it represents a circle:
  • The general form, \(Ax^2 + By^2 + Cy + D = 0\), identifies different conics.
  • In our case, \(3x^2 + 3y^2 - y = 0\) is a circle centered at the origin but slightly shifted due to the y-term.
  • Drawing its graph, one can see the symmetric circular shape.
Understanding conic sections is essential for recognizing and graphing the resultant form from coordinate conversions.

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Most popular questions from this chapter

Solve: \(\frac{5 x}{x+2}=\frac{x}{x-2}\)

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} \frac{1}{2} x+\frac{1}{3} y=3 \\ \frac{1}{4} x-\frac{2}{3} y=-1 \end{array}\right. $$

Solve each system of equations. If the system has no solution, state that it is inconsistent. Graph the lines of the system. $$ \left\\{\begin{array}{l} 5 x-y=21 \\ 2 x+3 y=-12 \end{array}\right. $$

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Simplify: \(\left(\frac{18 x^{4} y^{5}}{27 x^{3} y^{9}}\right)^{3}\)

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{aligned} 2 x-y &=-1 \\ x+\frac{1}{2} y &=\frac{3}{2} \end{aligned}\right. $$
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