Problem 142 - Calculus Volume 3

Question

Find rectangular coordinates for the given point in polar coordinates. $$ \left(2, \frac{5 \pi}{4}\right) $$

Step-by-Step Solution

Answer

Rectangular coordinates: \((-\sqrt{2}, -\sqrt{2})\).

Step 1: Understanding Polar Coordinates

Polar coordinates are defined as \((r, \theta)\), where \(r\) represents the radial distance from the origin, and \(\theta\) is the angle measured from the positive x-axis. Here, we're given the polar coordinates \((2, \frac{5\pi}{4})\), meaning the point is 2 units from the origin at an angle of \(\frac{5\pi}{4}\) radians.

Step 2: Converting Polar to Rectangular Coordinates

To convert to rectangular coordinates \((x, y)\), use the formulas: \(x = r \cos\theta\) and \(y = r \sin\theta\). Here, \(r = 2\) and \(\theta = \frac{5\pi}{4}\).

Step 3: Calculating the X-Coordinate

Compute \(x\) using the formula \(x = r \cos\theta\). Substitute both \(r\) and \(\theta\) to get \(x = 2 \cos\left(\frac{5\pi}{4}\right)\). Knowing that \(\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), calculate \(x = 2 \times -\frac{\sqrt{2}}{2} = -\sqrt{2}\).

Step 4: Calculating the Y-Coordinate

Compute \(y\) using the formula \(y = r \sin\theta\). Substitute both \(r\) and \(\theta\) to get \(y = 2 \sin\left(\frac{5\pi}{4}\right)\). Knowing that \(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}\), calculate \(y = 2 \times -\frac{\sqrt{2}}{2} = -\sqrt{2}\).

Step 5: Final Rectangular Coordinates

Combine the results from the previous steps to get the rectangular coordinates. The point \((x, y) = (-\sqrt{2}, -\sqrt{2})\) describes the location in rectangular coordinates.

Key Concepts

Polar Coordinates

Imagine standing at the center of a large circle with a compass in hand. This is similar to how polar coordinates work. In polar coordinates, every point in the plane can be described as a distance from the center (the origin) and an angle from a starting direction (usually the positive x-axis). The format is written as

• \((r, \theta)\)

where

• \(r\) is the radial distance from the origin,
• \(\theta\) is the angle measured in radians from the positive x-axis.

This is useful for describing motion or locations that naturally relate to circles, such as the path of a Ferris wheel. In the given exercise, the polar coordinates are \((2, \frac{5\pi}{4})\), meaning the point lies 2 units from the origin at an angle of \(\frac{5\pi}{4}\) radians, highlighting its position based on distance and direction.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use a grid system to pinpoint a location in a plane. Imagine a graph with horizontal and vertical lines intersecting at right angles. This is the foundation of rectangular coordinates, described as

• \((x, y)\)

where

• \(x\) represents the horizontal distance from the origin,
• \(y\) is the vertical distance from the origin.

These coordinates are highly intuitive because they resemble the grids used on graph paper, making it easy to visualize and plot points, lines, and shapes. The exercise converts the point from polar to rectangular coordinates, resulting in \((-\sqrt{2}, -\sqrt{2})\), a location specified by straightforward distances along the x and y axes.

Coordinate Conversion

Converting between polar and rectangular coordinates is a common task in mathematics, especially in trigonometry and calculus. This process allows flexibility in problem-solving, depending on which coordinate system better matches the context of a problem.
To convert from polar to rectangular coordinates, use the formulas:

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

where

• \(r\) is the distance from the origin (radius), and
• \(\theta\) is the angle in radians.

In the provided exercise, this conversion is applied to the polar coordinates \((2, \frac{5\pi}{4})\). By substituting these values into the conversion formulas, we determine both

• \(x = -\sqrt{2}\)
• \(y = -\sqrt{2}\)

This results in the rectangular coordinates \((-\sqrt{2}, -\sqrt{2})\), offering a convenient grid-based location description ideal for further mathematical computation or analysis.