Problem 19 - Calculus Early Transcendentals

Question

Express the following polar coordinates in Cartesian coordinates. \(\left(-4, \frac{3 \pi}{4}\right)\)

Step-by-Step Solution

Answer

Answer: The Cartesian coordinates equivalent to the given polar coordinates are \((2\sqrt{2}, -2\sqrt{2})\).

Step 1: Identify the given polar coordinates

The given polar coordinates are \((-4, \frac{3\pi}{4})\). Here, \(r = -4\) and \(\theta = \frac{3\pi}{4}\).

Step 2: Apply the conversion formulas

To convert the polar coordinates to Cartesian coordinates, we will use the following formulas: \(x = r \cos(\theta)\) \(y = r \sin(\theta)\)

Step 3: Find the value of x

Using the conversion formula for \(x\), we have: \(x = (-4) \cos\left(\frac{3\pi}{4}\right)\) The cosine of \(\frac{3\pi}{4}\) is \(-\frac{\sqrt{2}}{2}\). So, we end up with: \(x = (-4) \left(-\frac{\sqrt{2}}{2}\right) = 2\sqrt{2}\)

Step 4: Find the value of y

Using the conversion formula for \(y\), we have: \(y = (-4) \sin\left(\frac{3\pi}{4}\right)\) The sine of \(\frac{3\pi}{4}\) is \(\frac{\sqrt{2}}{2}\). So, we end up with: \(y = (-4) \left(\frac{\sqrt{2}}{2}\right) = -2\sqrt{2}\)

Step 5: Write the final Cartesian coordinates

Now that we have the \(x\) and \(y\) values, we can write the Cartesian coordinates as \((x, y)\) or \((2\sqrt{2}, -2\sqrt{2})\). So, the Cartesian coordinates equivalent to the given polar coordinates are: \((2\sqrt{2}, -2\sqrt{2})\)

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe the position of a point in a plane using two values:

• the distance from the origin (often represented as \(r\))
• the angle from the positive x-axis (represented by \(\theta\))

The pair \((r, \theta)\) is unique because it reflects how far and in which direction a point is, compared to the center of a coordinate system.
With the initial problem, the polar coordinates given are \((-4, \frac{3\pi}{4})\). This means:

• The point is 4 units away from the origin, but moving in the opposite direction since \(r\) is negative
• The direction or angle is \(\frac{3\pi}{4}\) radians from the positive x-axis

It's important to note the effects of using a negative \(r\), as it indicates the point is in the direction opposite of \(\theta\). This provides a concise and elegant way to convey information in navigation and physics problems.

Cartesian Coordinates

Cartesian coordinates are used to define a point in space through two coordinate axes intersecting at the origin:

• The x-coordinate tells how far along the horizontal axis the point is
• The y-coordinate informs how far along the vertical axis the point exists

For the provided exercise, the Cartesian coordinates were found to be \((2\sqrt{2}, -2\sqrt{2})\). This represents:

• An x-value of \(2\sqrt{2}\), indicating the point’s distance to the right of the origin
• A y-value of \(-2\sqrt{2}\), showing that the point is downward from the origin

This method is particularly useful in grid-like systems and is commonly used because of its straightforward rectangular layout, making it convenient for most mathematical and engineering calculations.

Coordinate Transformation

Coordinate transformation involves converting between different measures of describing a point's location, such as from polar to Cartesian coordinates. To accomplish this:

• The formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) are vital

These equations help translate the circular nature of polar coordinates into the linear framework of Cartesian coordinates. In the exercise's case:

• \(x = (-4) \cos(\frac{3\pi}{4}) = 2\sqrt{2}\)
• \(y = (-4) \sin(\frac{3\pi}{4}) = -2\sqrt{2}\)

By using trigonometric functions such as sine and cosine, the angle component of polar coordinates is mapped into direct x and y components.
This conversion process is critical in fields requiring precise physical placement, motion analysis, or solving complex equations where a coordinate transformation can simplify the task.