Problem 62 - 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. \(\left(\frac{9}{5}, \frac{11}{2}\right)\)

Step-by-Step Solution

Answer

The polar coordinates for the given point are approximately \((r, \theta)\) where \(r\) is the computed root value and \(\theta\) is the evaluated arctan value.

Step 1: Extract the rectangular coordinates

The rectangular coordinates given are: \(x = \frac{9}{5}\) and \(y = \frac{11}{2}\)

Step 2: Calculate the radius \(r\)

Using the formula \(r = \sqrt{{x}^2 + {y}^2}\), we can substitute the values of \(x\) and \(y\) to find \(r\). This results in \(r = \sqrt{{(\frac{9}{5})}^2 + {(\frac{11}{2})}^2}.\)

Step 3: Calculate the angle \(\theta\)

Using the formula \(\theta = \arctan(\frac{y}{x})\), substitute the values of \(x\) and \(y\) to find \(\theta\). This gives \(\theta = \arctan(\frac{\frac{11}{2}}{\frac{9}{5}})\). However, as the point is in the first quadrant, we don't need to adjust the value of \(\theta\).

Step 4: Evaluate \(r\) and \(\theta\)

Use a calculator or graphing utility to evaluate the values of \(r\) and \(\theta\). Make sure that the calculator is set to the correct mode (degrees or radians)

Key Concepts

Understanding Rectangular Coordinates

Rectangular coordinates, also called Cartesian coordinates, are used to specify the location of a point in a plane. They're expressed as \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position of a point. These coordinates are invaluable in various fields, such as engineering, physics, and computer graphics. To understand rectangular coordinates:
- Imagine a graph made up of two perpendicular lines, one horizontal (x-axis) and one vertical (y-axis).
- The point where these two lines intersect is called the origin, designated by \((0,0)\).
- Each point on the plane is defined by how far along the x-axis and y-axis it is from the origin.
For example, in the exercise, the rectangular coordinates are \(\left(\frac{9}{5}, \frac{11}{2}\right)\), meaning the point is positive along both axes and located in the first quadrant. Understanding this helps in visualizing the transformation into polar coordinates.

Using a Graphing Utility

Graphing utilities are tools, either software or calculators, that allow us to visualize mathematical equations and concepts. They are extremely useful when converting between coordinate systems. Here’s how they can help:

• Visualize complex numbers and functions by plotting them on a plane.
• Provide an easy way to calculate angles and distances by giving graphical representation.
• Allow us to set preferences, such as viewing angles in either degrees or radians based on the problem's requirements.

A graphing utility transitions theoretical calculations to practical, visual insights, making it essential for verifying polar coordinates derived from rectangular ones. Students can input rectangular coordinates and instantly see their position and corresponding polar coordinates on the utility's graph.

Radius Calculation in Polar Coordinates

The radius in polar coordinates, denoted as \(r\), represents the distance from the origin to the point in the plane. To calculate \(r\), you apply the Pythagorean theorem in the context of the x and y axes. The formula is \(r = \sqrt{x^2 + y^2}\). For example, given \(x = \frac{9}{5}\) and \(y = \frac{11}{2}\):
- Calculate the square of each value: \(\left(\frac{9}{5}\right)^2\) and \(\left(\frac{11}{2}\right)^2\).
- Add these squares together: \(\frac{81}{25} + \frac{121}{4}\).
Ultimately, take the square root to find the radius \(r\). Calculating \(r\) gives you a key component of the polar coordinates, indicating how far the point lies from the origin.

Angle Calculation in Polar Coordinates

The angle \(\theta\) in polar coordinates indicates the direction of the point concerning the positive x-axis. It’s calculated using the tangent function and its inverse, the arctangent. For our exercise, the formula is \(\theta = \arctan\left(\frac{y}{x}\right)\). Here's how the calculation proceeds:
- Divide \(y\) by \(x\): \(\frac{\frac{11}{2}}{\frac{9}{5}}\).
- Calculate the arctangent of this value to determine \(\theta\).
Because the point \(\left(\frac{9}{5}, \frac{11}{2}\right)\) is in the first quadrant (both coordinates are positive), \(\theta\) needs no adjustment. This calculation ensures the angle accurately reflects the point's direction on the graph. Calculators and graphing utilities should be set to either radians or degrees, as required.