Problem 20 - Precalculus with Calculus Previews

Question

Find the rectangular coordinates for each point with the given polar coordinates. $$ (-5, \pi / 2) $$

Step-by-Step Solution

Answer

The rectangular coordinates are (0, -5).

Step 1: Understand Polar Coordinates

Polar coordinates are expressed as \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis. Here, the given polar coordinates are \((-5, \pi / 2)\).

Step 2: Recognize Negative Radius

The value \(r = -5\) indicates that the point is 5 units away from the pole, but in the opposite direction of the angle given. We need to account for this when converting to rectangular coordinates.

Step 3: Convert Angle to Rectangular Adjustments

The angle \(\theta = \pi / 2\) is equivalent to 90 degrees and points directly along the positive y-axis. However, with a negative radius, the point would be along the negative y-axis.

Step 4: Calculate Rectangular Coordinates

Rectangular coordinates \((x, y)\) can be calculated using the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). With \(r = -5\) and \(\theta = \pi / 2\), we find: \[ x = -5 \cos(\pi / 2) = -5 \cdot 0 = 0 \] \[ y = -5 \sin(\pi / 2) = -5 \cdot 1 = -5 \]. Thus, the rectangular coordinates are \((0, -5)\).

Key Concepts

Polar Coordinates

Polar coordinates are an alternative to the typical rectangular (or Cartesian) coordinate system which uses an x and y-axis. Unlike rectangular coordinates, polar coordinates describe a point based on its distance from a central point (the pole) and the angle from a reference direction (usually the positive x-axis).
This system is particularly useful in situations where relationships and directions are more naturally described in terms of angles and distances rather than horizontal and vertical displacements.

• The first value, \( r \), is the radial coordinate representing the distance from the origin to the point.
• The second value, \( \theta \), is the angular coordinate and often measured in radians.

In this context, the point \((-5, \pi / 2)\) uses a negative radius, which brings another layer of consideration into the mix that we'll explore further in this article.

Coordinate Conversion

Converting polar coordinates into rectangular coordinates is a common task in mathematics. To perform this conversion, we use trigonometric formulas involving \( \cos \) and \( \sin \). This conversion process allows us to express a given point using the familiar rectangular coordinate format, \((x, y)\).
The conversion is done using these key formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)

These formulas rely on understanding that the angle \( \theta \) determines the direction of the point, and the radial distance \( r \) tells us how far along this direction the point lies.
For instance, converting the polar coordinates \((-5, \pi / 2)\), involves calculating \( x \) and \( y \) using the specified radius and angle.

Trigonometric Functions

Trigonometric functions, specifically cosine \( \cos \) and sine \( \sin \), play a pivotal role in the process of coordinate conversion. These functions relate angles directly to the ratios of the sides of right triangles, which can then be applied to polar coordinates in order to convert them to rectangular coordinates.
When you have the angle \( \theta = \pi/2 \), such as in our problem:

• \( \cos(\pi/2) = 0 \)
• \( \sin(\pi/2) = 1 \)

These particular values simplify the conversion process significantly.
For instance, integrating these into our conversion formulas with \( r = -5 \), the calculation for the x-coordinate results in \( x = -5 \times 0 = 0 \), and for the y-coordinate results in \( y = -5 \times 1 = -5 \). These are critical steps that yield the rectangular coordinates \((0, -5)\).

Negative Radius

Understanding a negative radius in polar coordinates can sometimes be tricky. Geometrically, a negative radius can be thought of as the opposite direction of the specified angle \( \theta \). If \( r \) is negative, the point actually lies the same distance from the origin, but in the direction directly opposite the angle.
When you encounter a point such as \((-5, \pi/2)\), the negative radius implies the point is below the x-axis despite the angle directing us upwards.

• This concept means shifting the intended point 180 degrees from the angle.
• Visualize it as extending a line from the origin; a negative radius reverses this line through the origin to the opposite side.

Recognizing and correctly handling the sign of \( r \) is essential in ensuring accurate conversions, which here results in the correct rectangular coordinates of \((0, -5)\).