Problem 146 - Calculus Volume 3

Question

Find rectangular coordinates for the given point in polar coordinates. $$ \left(-3, \frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

Rectangular coordinates are \(\left( \frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2} \right)\).

Step 1: Understanding Polar Coordinates

Polar coordinates are given in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle from the positive x-axis, measured in radians counterclockwise.

Step 2: Recognize Given Values

In this problem, the polar coordinates are \((-3, \frac{3\pi}{4})\). Here, \(r = -3\) and \(\theta = \frac{3\pi}{4}\).

Step 3: Use Conversion Formulas

The conversion from polar to rectangular coordinates is done using the formulas: \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 4: Calculate x-coordinate

Substitute the values into the formula to find \(x\):\[x = -3 \cos\left(\frac{3\pi}{4}\right)\]\[x = -3 \times \left(-\frac{\sqrt{2}}{2}\right)\]\[x = \frac{3\sqrt{2}}{2}\]

Step 5: Calculate y-coordinate

Substitute the values into the formula to find \(y\):\[y = -3 \sin\left(\frac{3\pi}{4}\right)\]\[y = -3 \times \frac{\sqrt{2}}{2}\]\[y = -\frac{3\sqrt{2}}{2}\]

Step 6: Consolidate Rectangular Coordinates

After calculating separately, we find that the rectangular coordinates corresponding to the polar coordinates \((-3, \frac{3\pi}{4})\) are:\[\left( \frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2} \right)\]

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing a point in a plane by describing its distance from a fixed point and the angle from a fixed direction. Here’s how it works:

• Radial Distance (\( r \)): This is the straight-line distance from the point to the origin (0,0).
• Angle (\( \theta \)): This indicates the direction, measured in radians from the positive x-axis.

Unlike Cartesian coordinates, which are noted as \((x, y)\), polar coordinates are usually displayed as \((r, \theta)\). It is important to understand that \( r \) can be negative. A negative radial distance means the point is in the opposite direction of the angle \( \theta \).
In the given exercise, the point \((-3, \frac{3\pi}{4})\) has a negative radial distance with an angle of \( \frac{3\pi}{4} \) radians, which means we move \(3\) units in the direction opposite to that indicated by the angle \( \frac{3\pi}{4} \). This changes how the point is positioned on the plane based on traditional polar coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points with their horizontal and vertical distances from the origin. These are noted as \((x, y)\). This system is based on the coordinate grid, where each point is described by how far it is along the x-axis and y-axis.
To convert from polar coordinates to rectangular coordinates, you use the formulas:

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

In this particular exercise, by inserting the polar coordinates \((-3, \frac{3\pi}{4})\) into the formulas, we calculate:

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

Thus, the corresponding rectangular coordinates are \( \left( \frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2} \right) \). These calculations show how polar coordinates translate into a familiar x and y grid system. Understanding these formulas and their use is crucial to successfully transitioning between the two coordinate systems.

Trigonometric Functions

Trigonometric functions are the bridge between polar and rectangular coordinates. They help to relate the angle and the radial distance in polar coordinates to the horizontal and vertical distances in the rectangular system.
The key trigonometric functions involved in this conversion are sine and cosine:

• Sine (\( \sin \)): Calculates the y-component of a point on the unit circle based on an angle.
• Cosine (\( \cos \)): Determines the x-component of a point on the unit circle for a given angle.

For the angle \( \frac{3\pi}{4} \), these functions have well-known values:

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

Applying these to the polar point \((-3, \frac{3\pi}{4})\), we use cosine to find the x-coordinate and sine for the y-coordinate. Understanding these functions and their properties, especially with known angles, allows you to perform the conversion between polar and rectangular coordinates seamlessly.