Problem 29 - Precalculus with Limits

Question

In Exercises 29-36, use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. \(\left(2, 2\pi/9\right)\)

Step-by-Step Solution

Answer

The rectangular coordinates for the given polar coordinates \((2, \frac{2\pi}{9})\) are approximately \((x,y)\) where x and y are the calculated values rounded to two decimal points.

Step 1: Identify the given coordinates

The given polar coordinates are \(r = 2\) and \(\theta = \frac{2\pi}{9}\).

Step 2: Apply the conversion formula

The formulas to convert polar coordinates into rectangular coordinates are given by \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). In this case, it will process to \(x = 2\cos(\frac{2\pi}{9})\) and \(y = 2\sin(\frac{2\pi}{9})\).

Step 3: Use a graphing utility

Using a graphing utility or calculator, calculate the values of x and y using the formulas obtained in the previous step.

Step 4: Round the results

Round the calculated values to two decimal places.

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe the location of points in a plane. Unlike rectangular coordinates, which use an x-axis and y-axis, polar coordinates use a radius and an angle.

• The radius, denoted as \(r\), is the distance from the origin to the point.
• The angle, \(\theta\), is measured from the positive x-axis to the line segment that connects the origin to the point.

This system is particularly useful in scenarios involving circular or spiral patterns, where specifying a location is simpler in terms of angle and distance rather than horizontal and vertical offsets. Polar coordinates are expressed as \((r, \theta)\), making them an excellent choice for problems involving curves and cycles. Understanding these concepts is crucial for translating between different coordinate systems and effectively tackling exercises like the one given.

Graphing Utility

A graphing utility, or graphing calculator, is a powerful tool that aids in visualizing and computing complex mathematical tasks. For the problem of converting polar coordinates to rectangular coordinates, it comes in handy by:

• Quickly computing the trigonometric functions required, such as cosine and sine.
• Providing a visual representation of the coordinates, making it easier to understand the relationships between the points.

For example, in converting the polar coordinates \((2, \frac{2\pi}{9})\), a graphing utility rapidly performs the calculations: \[\begin{align*} x &= 2 \cdot \cos\left(\frac{2\pi}{9}\right), \ y &= 2 \cdot \sin\left(\frac{2\pi}{9}\right)\end{align*}\]These tools are especially beneficial in educational settings, allowing students to focus on understanding concepts rather than becoming bogged down with manual calculations.

Conversion Formulas

Converting between polar and rectangular coordinates is a fundamental part of working with two-dimensional coordinate systems. The conversion formulas are derived from basic trigonometric properties:

• For x-coordinate: \(x = r\cos(\theta)\)
• For y-coordinate: \(y = r\sin(\theta)\)

These formulas allow you to translate from the circular perspective of polar coordinates to the linear perspective of rectangular coordinates. When dealing with \(r = 2\) and \(\theta = \frac{2\pi}{9}\), using these formulas helps you find precise values for x and y, notably:\[x = 2 \cdot \cos\left(\frac{2\pi}{9}\right), \y = 2 \cdot \sin\left(\frac{2\pi}{9}\right)\]Understanding these conversion formulas is key in switching between different ways of depicting points on a plane, and they are widely used in navigation, physics, and engineering fields.