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Chapter 8: Problem 5

Convert the Cartesian coordinates to polar coordinates. $$(1,1)$$

Short Answer

Expert verified
The polar coordinates are \( \left(\sqrt{2}, \frac{\pi}{4}\right) \).

Step by step solution

01

Identify the Cartesian Coordinates

The given Cartesian coordinates are \( x = 1 \) and \( y = 1 \). These will be used in our conversion formulas.
02

Calculate the Radius

The radius \( r \) in polar coordinates is calculated as: \[ r = \sqrt{x^2 + y^2} \]Substitute the values: \[ r = \sqrt{1^2 + 1^2} = \sqrt{2} \]
03

Calculate the Angle

The angle \( \theta \) can be found using the formula: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]Substitute the values: \[ \theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \] Since both \( x \) and \( y \) are positive, the point lies in the first quadrant, so no adjustment to \( \theta \) is needed.
04

Write the Polar Coordinates

Now that we have \( r = \sqrt{2} \) and \( \theta = \frac{\pi}{4} \), we write the polar coordinates as: \[ (r, \theta) = \left(\sqrt{2}, \frac{\pi}{4}\right) \]

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Key Concepts

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

Cartesian Coordinates
Cartesian coordinates are a way to determine the exact position of a point in a two-dimensional plane. In this system, every point is described by a pair of numbers \((x, y)\). Here, \(x\) represents the horizontal distance from the origin (0,0), and \(y\) represents the vertical distance. For example, the point \((1, 1)\) tells us that we move one unit right and one unit up from the origin to locate the point.
  • The x-coordinate shows how far left or right the point is.
  • The y-coordinate shows how far up or down it is.
This straightforward coordinate system makes it easy to represent and visualize points on the plane.
Radius Calculation
The radius in polar coordinates describes how far a point is from the origin. To find it, you use the formula for distance in the Cartesian plane:
\[ r = \sqrt{x^2 + y^2} \] This is derived from the Pythagorean theorem, which can compute the distance from the origin to any point鈥攆orming a right triangle will help visualize this.
  • Calculate squared values of both x and y.
  • Add these squared values together.
  • Take the square root of this sum to find the radius \(r\).
So, for the point \((1, 1)\), the radius calculation is \( \sqrt{1^2 + 1^2} = \sqrt{2} \). This value tells us the direct line distance from the point to the origin.
Angle Calculation
Determining the angle \(\theta\) in polar coordinates helps you know the direction of the point relating to the positive x-axis. You use the inverse tangent function \(\tan^{-1}\), which requires the ratio of the y and x coordinates:
\[ \theta = \tan^{-1}\left( \frac{y}{x} \right) \] This gives an angle in radians, the standard unit in polar coordinates. To do this:
  • Divide the y-coordinate by the x-coordinate.
  • Use the \(\tan^{-1}\) function to find the angle \(\theta\).
For our example \((1, 1)\), the angle calculation results in \( \tan^{-1}(1) = \frac{\pi}{4} \). This angle indicates the direction of the point from the origin, typically given in radians.
Coordinate Transformation
Converting from Cartesian to polar coordinates translates position information into a different frame of reference. This transformation is used in navigation, physics, and engineering.
To transform the Cartesian coordinates \((x, y)\) to polar \((r, \theta)\):
  • Calculate the radius \(r\) as \(\sqrt{x^2 + y^2}\).
  • Determine the angle \(\theta\) as the inverse tangent of \(\frac{y}{x}\).
Together, these values \(r\) and \(\theta\) provide a full description of the point in polar terms. For the given \((1, 1)\), we found \(r = \sqrt{2}\) and \(\theta = \frac{\pi}{4}\). This transformation allows us to understand not just where a point is, but also its direction from the origin.

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Most popular questions from this chapter

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