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Chapter 6: Problem 23

The rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in \((0,2 \pi]\). Round to three decimal places. $$ (2,2) $$

Short Answer

Expert verified
The polar coordinates are approximately (2.828, 0.785) and (2.828, 7.068).

Step by step solution

01

Identify Coordinates

The given rectangular coordinates of the point are (2, 2). We need to convert these into polar coordinates, which consist of a radius \( r \) and an angle \( \theta \).
02

Calculate the Radius

The radius \( r \) can be found using the formula \( r = \sqrt{x^2 + y^2} \), where \( x = 2 \) and \( y = 2 \). Thus, \( r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828 \).
03

Determine the Angle from Components

The angle \( \theta \) can be found using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \). Here, \( y = 2 \) and \( x = 2 \), so \( \theta = \tan^{-1}(1) = \frac{\pi}{4} \approx 0.785 \) radians.
04

Calculate Second Angle Option

In polar coordinates, an angle can also be expressed by adding \( 2\pi \) to any angle, as it results in the same direction. Thus, a second set of polar coordinates is obtained by adding \( 2\pi \) to \( \frac{\pi}{4} \), giving us \( \theta = \frac{\pi}{4} + 2\pi \approx 7.068 \).
05

Assemble the Polar Coordinates

The two possible sets of polar coordinates are \( (2\sqrt{2}, \frac{\pi}{4}) \) or approximately \( (2.828, 0.785) \) and \( (2\sqrt{2}, \frac{\pi}{4} + 2\pi) \) or approximately \( (2.828, 7.068) \).

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates, also known as Cartesian coordinates, represent the position of a point in a plane using an ordered pair \(x, y\). In this coordinate system:
  • The \(x\) value indicates the horizontal position.
  • The \(y\) value indicates the vertical position.
For the point \( (2, 2) \), both the \(x\) and \(y\) values are 2. This means the point is situated two units to the right and two units up from the origin \( (0,0) \) on the Cartesian plane.

The concept of converting rectangular coordinates into polar coordinates involves translating the position from the usual grid system to one based on radius and angle, which is often useful in different fields like physics and engineering.
Radius Calculation
The first step in converting rectangular coordinates \( (x, y) \) to polar coordinates \( (r, \theta) \) is calculating the radius, \( r\). The radius is the distance from the origin to the point and can be found using the formula:
  • \( r = \sqrt{x^2 + y^2} \)
For our point \( (2, 2) \):
  • \( r = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8}\)
  • \( \sqrt{8} = 2\sqrt{2} \approx 2.828 \)
Here, \( 2\sqrt{2} \) or approximately 2.828 is the Euclidean distance from the origin to the point, representing the radius in polar coordinates.

This distance helps us understand how far the point is from the center of our coordinate system, allowing for a different kind of measurement of position beyond simple horizontal and vertical displacement.
Angle Determination
Once we have the radius, the next crucial step is determining the angle \( \theta \), to complete converting to polar coordinates. The angle represents the direction of the point from the origin and is measured in radians.

To calculate \( \theta \), use the equation:
  • \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \)
For the point \( (2, 2) \):
  • \( \frac{y}{x} = \frac{2}{2} = 1 \)
  • \( \theta = \tan^{-1}(1) = \frac{\pi}{4} \approx 0.785 \ ext{radians} \)
Remember, angles in polar coordinates can be expressed in multiple ways due to the periodic nature of circular measurement:
  • Adding \( 2\pi \) to any angle indicates the same direction.
  • Thus, another angle option is \( \theta = \frac{\pi}{4} + 2\pi \), which simplifies approximately to \( 7.068 \ ext{radians} \).
These angle calculations allow you to fully define the point's position in the polar coordinate system.

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Most popular questions from this chapter

For the following exercises, sketch the graph of each conic. $$ y^{2}=20 x $$

For the following exercises, consider the following polar equations of conics. Determine the eccentricity and identify the conic. $$ r=\frac{3}{413 \sin \theta} $$

For the following exercises, sketch the graph of each conic. $$ \frac{x^{2}}{16}-\frac{y^{2}}{9}=1 $$

For the following exercises, sketch the parametric curve and eliminate the parameter to find the Cartesian equation of the curve. $$ x=\sin \theta, y=1-\csc \theta, 0 \leq \theta \leq 2 \pi $$

Find the slope of the tangent line to the given polar curve at the point given by the value of \(\theta\).\(r=3 \cos \theta, \theta=\frac{\pi}{3}\)
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