Problem 9 - Thomas Calculus

Question

Find the polar coordinates, \(0 \leq \theta<2 \pi\) and \(r \leq 0,\) of the following points given in Cartesian coordinates. $$\begin{array}{ll}{\text { a. }(3,3)} & {\text { b. }(-1,0)} \\ {\text { c. }} {(-1, \sqrt{3})} & {\text { d. }(4,-3)}\end{array}$$

Step-by-Step Solution

Answer

(3,3): \((3\sqrt{2}, \pi/4)\), (-1,0): \((1, \pi)\), (-1, √3): \((2, 2\pi/3)\), and (4,-3): \((5, 5.64)\)."

Step 1: Convert Cartesian to Polar Coordinates [Point (3,3)]

To find the polar coordinates \((r, \theta)\) for the Cartesian point \((3,3)\), we use the following formulas: 1. Calculate the radius \(r\) using \(r = \sqrt{x^2 + y^2}\). \[ r = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \]2. Find the angle \(\theta\) using \(\theta = \arctan\left(\frac{y}{x}\right)\). \[ \theta = \arctan\left(\frac{3}{3}\right) = \arctan(1) = \frac{\pi}{4} \]So, the polar coordinates are \((3\sqrt{2}, \frac{\pi}{4})\).

Step 2: Convert Cartesian to Polar Coordinates [Point (-1,0)]

To find the polar coordinates for the point \((-1,0)\):1. Calculate the radius \(r\): \[ r = \sqrt{(-1)^2 + 0^2} = 1 \]2. Determine the angle \(\theta\): Since the point lies on the negative x-axis, \(\theta = \pi\).Thus, the polar coordinates are \((1, \pi)\).

Step 3: Convert Cartesian to Polar Coordinates [Point (-1, √3)]

For the point \((-1, \sqrt{3})\):1. Calculate \(r\): \[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2 \]2. Find \(\theta\) using \(\theta = \arctan\left(\frac{y}{x}\right)\): \[ \theta = \arctan\left(\frac{\sqrt{3}}{-1}\right) \] This corresponds to an angle of \(\frac{2\pi}{3}\) in the second quadrant.Hence, the polar coordinates are \((2, \frac{2\pi}{3})\).

Step 4: Convert Cartesian to Polar Coordinates [Point (4,-3)]

For the point \((4,-3)\):1. Calculate \(r\): \[ r = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5 \]2. Find \(\theta\): \[ \theta = \arctan\left(\frac{-3}{4}\right) \] The angle calculated as \(\arctan\left(\frac{-3}{4}\right)\) needs to be adjusted to ensure \(\theta\) is in the right quadrant (fourth quadrant). So, \(\theta = 2\pi - \arctan\left(\frac{3}{4}\right)\), approximately \(5.64\) radians.The polar coordinates are \((5, 5.64)\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a system that allows us to pinpoint a location in a two-dimensional plane using two values: an x-coordinate and a y-coordinate.

• The x-coordinate tells us how far to move along the horizontal axis.
• The y-coordinate indicates how far to move along the vertical axis.

This system is widely used because it is straightforward and aligns with how we naturally perceive space. For example, a point (3,3) is located three units to the right and three units up from the origin, where both the x and y axes meet. In the Cartesian plane, each point is unique, defined by the two coordinates \( (x,y) \). This is why we tend to use this method of representation in various fields, such as graphing equations or plotting data points in statistics.

Coordinate Conversion

Coordinate conversion involves transforming a point represented in one coordinate system into another. In our case, we are converting Cartesian coordinates \((x,y)\) into polar coordinates \((r, \theta)\). This transformation helps us understand a point's location using distance and angle, rather than horizontal and vertical movement.Conversion uses specific formulas for

• Radius \( r = \sqrt{x^2 + y^2} \), which measures the distance from the origin to the point.
• Angle \( \theta = \arctan (\frac{y}{x}) \), which tells us the direction around a circle, from the positive x-axis.

This conversion is particularly useful in fields where circular symmetry and periodic functions are prevalent, such as in trigonometry or when dealing with rotating bodies in physics.

Angle Calculation

Calculating the angle, \( \theta \), is an essential part of converting Cartesian coordinates to polar coordinates. It tells us the direction in which a point lies relative to the positive x-axis. We find \( \theta \) using the formula:\[ \theta = \arctan \left(\frac{y}{x}\right) \]This calculation gives an angle in radians, ensuring it fits within the range \( 0 \leq \theta < 2\pi \). It's important to consider the quadrant in which the point lies, to adjust the angle correctly. While \( \arctan \) initially provides an angle assuming the point is in the first or fourth quadrant, you might need to adjust it:

• Second quadrant: add \( \pi \).
• Third quadrant: add \( \pi \).
• Fourth quadrant: use \( 2\pi - \text{angle}\) for directions below the x-axis.

Radius Calculation

The radius \( r \) is a measure of how far a point is from the origin, \( (0,0) \), in the polar coordinate system. It is calculated using the Pythagorean theorem as:\[ r = \sqrt{x^2 + y^2} \]This formula essentially computes the hypotenuse of the right triangle formed by the point and the axes. The radius will always be a non-negative number.For example, for the point \((3,3)\), we calculate:\[ r = \sqrt{3^2 + 3^2} = \sqrt{18} = 3\sqrt{2} \]In practice, this value provides important insight into how far the point is from the origin, effectively combining x and y displacements into a single measure. Calculating \( r \) is crucial for applications involving waves, orbits, and various geometry problems.