Problem 39 - Precalculus

Question

Use a graphing utility to find the rectangular coordinates of the point given in polar coordinates. Round your results to two decimal places. $$(2.5,1.58)$$

Step-by-Step Solution

Answer

The rectangular coordinates of the given point in polar coordinates are approximately \((-0.28, 2.44)\)

Step 1: Identify given coordinates

The polar coordinates given are \((r, \theta) = (2.5, 1.58)\)

Step 2: Use the conversion formula for x-coordinate

Calculate the \(x\) coordinate by using the formula \(x = r\cos(\theta)\). Therefore, \(x = 2.5\cos(1.58)\)

Step 3: Calculate the x coordinate

After calculating, we find \(x \approx -0.28\) rounded to two decimal places.

Step 4: Use the conversion formula for y-coordinate

Now calculate the \(y\) coordinate using the formula \(y = r\sin(\theta)\). Thus, \(y = 2.5 \sin(1.58)\)

Step 5: Calculate the y coordinate

After calculating, we find \(y \approx 2.44\) rounded to two decimal places.

Key Concepts

Graphing Utility

In geometry and trigonometry, the process of visualizing coordinate systems is crucial to understanding the positioning of points in a space. A graphing utility is an essential tool for students to explore and visualize these concepts. Modern graphing utilities go beyond simple plotting; they can convert coordinates between different systems, zoom in on areas of interest, or even animate the effects of transformations.

When tasked with finding rectangular coordinates from polar coordinates, a graphing utility allows for a more interactive approach. First, it would visually illustrate the polar point, facilitating comprehension of its location. Then, by employing the utility’s ability to plot the corresponding rectangular point, students can engage with the concept, witnessing the transformation right on their screens. Ultimately, this interaction improves retention and enhances understanding.

Coordinate Conversion

The translation of points between coordinate conversion systems is a fundamental aspect of mathematics, particularly when dealing with curves and shapes. Polar and rectangular coordinates describe positions in two different ways. Polar coordinates use a radius and an angle from the origin to specify a point, while rectangular coordinates use two perpendicular displacements along the x and y axes.

To convert from polar to rectangular coordinates, two simple equations are used: for the x-coordinate, calculate \(x = r\cos(\theta)\), and for the y-coordinate, use \(y = r\sin(\theta)\). These equations stem from the trigonometric relationships found within a right-angled triangle, with the polar radius as the hypotenuse. It’s essential for students to internalize these conversions as they are not only key in mathematics but also in fields like physics, engineering, and computer science, where different coordinate systems are used depending on the context.

Rounding Decimal Places

In mathematical computations, especially those involving real-world data or measurements, rounding decimal places is a vital skill. Rounding helps to simplify numbers, making them easier to work with, and often is necessary to match the precision dictated by the context of a problem.

When converting coordinates from polar to rectangular form, the final step often involves rounding the results to a specified number of decimal places. Rounding to two decimal places, as the exercise requires, means adjusting the calculated value to the nearest hundredth. It's important to note that if the digit in the third decimal place is 5 or more, the second decimal place is rounded up; otherwise, it remains the same. Mastering rounding ensures the accuracy of solutions and helps conform to the requirements of different scenarios students might encounter academically or professionally.