Problem 60 - Precalculus with Limits

Question

In Exercises 55-64, use a graphing utility to find one set of polar coordinates for the point given in rectangular coordinates. \(\left(5, -\sqrt{2}\right)\)

Step-by-Step Solution

Answer

The point (5, -\( \sqrt{2} \)) in rectangular coordinates corresponds to about (5.1, -0.28) in polar coordinates.

Step 1: Compute the radial coordinate

First we need to compute the radial coordinate (r) which is the distance from the origin to the point. The formula is \(r = \sqrt{x^2 + y^2}\). Here, x is 5 and y is \(-\sqrt{2}\). Therefore, \(r = \sqrt{(5^2 + (-\sqrt2)^2)}\).

Step 2: Compute the angular coordinate

Next, we compute the angular coordinate (θ), which is the angle measured in the positive x direction. We can find θ using the formula \(θ = \arctan(\frac{y}{x})\). Here, y = \(-\sqrt{2}\) and x = 5. So, \(θ = \arctan(\frac{-\sqrt{2}}{5})\).

Step 3: Adjust the angular coordinate

Since we have y < 0 while x > 0, the point is in the 4th Quadrant. In the context of the polar coordinate, this specifies that the terminal side of the angle measure, θ, is in the 4th Quadrant. However, the arctangent function normally gives values between -π/2 and π/2 (-90 and 90 degrees) which corresponds to Quadrant I and IV. As we get the positive value for arctan calculation and the point is in Quadrant IV, this means the arctan value is indeed the actual angle θ.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to locate points on a plane using two values. These values are typically labeled as x and y.

• The x-coordinate tells you how far to move horizontally from the origin.
• The y-coordinate tells you how far to move vertically.

For example, the point (5, -\(\sqrt{2}\)) means you move 5 units to the right (as x is positive) and \(\sqrt{2}\) units down (as y is negative) from the origin. Understanding this system is critical as it is the foundation for identifying positions on the graph before converting them to other forms like polar coordinates. It's important to visualize this step as placing a point on a grid with the origin at the center.

Graphing Utility

A graphing utility is a tool, like a graphing calculator or software, that helps plot functions and points on a coordinate plane. It is especially helpful when you need to visualize mathematical concepts or verify solutions. When working with polar and rectangular coordinates, a graphing utility can:

• Show you how a point in rectangular form translates to polar form.
• Provide a graphical representation of angles and distances.
• Draw complex shapes and analyze the position and behavior of these shapes thoroughly.

Having a visual aid can help solidify your understanding of how points and graphs interact in different coordinate systems.

Quadrants

In mathematics, a coordinate plane is divided into four areas known as quadrants. Each quadrant is defined by the signs of the x and y coordinates:

• Quadrant I - Both x and y are positive.
• Quadrant II - x is negative, y is positive.
• Quadrant III - Both x and y are negative.
• Quadrant IV - x is positive, y is negative.

Understanding which quadrant a point belongs to helps determine the appropriate angle when converting rectangular coordinates to polar coordinates. For the point (5, -\(\sqrt{2}\)), it is in Quadrant IV since x is positive and y is negative, indicating an angle measured in the positive x direction.

Arctangent Function

The arctangent function, denoted as \(\arctan\), is used to find the angle whose tangent is a given number. It is essential in converting rectangular coordinates to polar coordinates because it helps determine the angle \(\theta\).In general, \(\arctan(y/x)\) gives the angle \(\theta\) whose tangent is \(\frac{y}{x}\). However, arctangent values range from \(-\pi/2\) to \(\pi/2\), which covers just the 1st and 4th quadrants.For our example, obtaining \(\theta = \arctan\left(\frac{-\sqrt{2}}{5}\right)\) indicates the angle lies in the 4th quadrant because y is negative while x is positive. This use of arctangent ensures we establish the correct angle measure consistently with the coordinate's quadrant location.