Problem 37 - 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. $$(-4.5,1.3)$$

Step-by-Step Solution

Answer

To obtain the rectangular coordinates, the formulas \(x = r*cos(\theta)\) and \(y = r*sin(\theta)\) should be used with the provided polar coordinates. The result should be validated by converting the computed rectangular coordinates back into polar form, to ensure consistency with the provided polar coordinates.

Step 1: Understand the polar coordinates

Polar coordinates are denoted as (r,\theta), where r signifies the distance from the origin to the point and \theta stands for the angle from the positive x-axis to the point. The given polar coordinates are (-4.5, 1.3). Here, -4.5 (r) is the distance from the origin to the point and 1.3 (\theta) is the angle.

Step 2: Apply the conversion formula

We need to apply the conversion formulas for polar to rectangular coordinates. The x-coordinate can be found by multiplying the distance (r) by the cosine of the angle (\theta): \(x = r*cos(\theta)\)\nSimilarly, the y-coordinate can be found by multiplying the distance (r) by the sine of the angle (\theta): \(y = r*sin(\theta)\)

Step 3: Compute the rectangular coordinates

When the given r and \theta values are substituted into the formulas, we get: \(x = -4.5*cos(1.3)\) and \(y = -4.5*sin(1.3)\). By calculating these expressions, the rectangular coordinates can be found.

Step 4: Validate the result

Check the calculated coordinates by converting them back into polar coordinates using a graphing utility or the formulae. This should match the original given polar coordinates.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional way to describe a point's position, using a combination of distance and angle from a specific origin point. In this system:

• r: Represents the radius or the distance from the origin to the point.
• \(\theta\): Denotes the angle measured in radians from the positive x-axis to the line connecting the origin to the point.

For example, in the given polar coordinates \((-4.5, 1.3)\), the radius \(r\) is \(-4.5\). Here, \(\theta\) is \(1.3\) radians, indicating the point's direction from the origin. It's important to note that when \(r\) is negative, it indicates the point is in the opposite direction of the angle \(\theta\).

Rectangular Coordinates

Rectangular coordinates, often referred to as Cartesian coordinates, are another system of defining a point's position in space. Unlike polar coordinates, which use a distance and angle, rectangular coordinates use a pair of perpendicular lines: a horizontal (x-axis) and a vertical (y-axis). Each point is represented by:

• x-coordinate: The horizontal distance from the origin.
• y-coordinate: The vertical distance from the origin.

The transformation from polar to rectangular involves converting the radial distance and angle of the point into how far left/right and up/down it is from the origin. This shift in description can make graphing equations or plotting data more intuitive in some cases.

Conversion Formulas

To switch from polar to rectangular coordinates, we use specific mathematical formulas that help translate position between these different systems:

• The x-coordinate can be found with the formula: \(x = r\cos(\theta)\)
• The y-coordinate can be found with the formula: \(y = r\sin(\theta)\)

These formulas are derived from the trigonometric relationships in a right triangle. The angle \(\theta\) determines the proportions of the triangle's sides, with \(\cos(\theta)\) and \(\sin(\theta)\) representing the horizontal and vertical components, respectively.
In our example, using \(r = -4.5\) and \(\theta = 1.3\), you'll calculate the coordinates \(x\) and \(y\) by assessing the cosine and sine of \(1.3\), then multiplying these by \(-4.5\). This process will yield the final rectangular coordinates.

Graphing Utility

A graphing utility can be a valuable tool for visualizing and verifying coordinate transformations. It allows you to input polar coordinates and obtain the corresponding rectangular coordinates without doing manual calculations.
Such utilities often come in the form of:

• Graphing calculators
• Software applications
• Online graphing tools

Using a graphing utility ensures accuracy in conversion and provides a visual representation of the point. It can also help confirm results by allowing you to work backward: input the found rectangular coordinates and convert them back into polar to ensure consistency. This way, students can both see and understand the transformation process in action.