Problem 52 - Algebra and Trigonometry

Question

Polar coordinates of a point are given. Find the rectangular coordinates of each point. $$ \left(-6,-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((-3\sqrt{2}, 3\sqrt{2})\).

Step 1 - Define the given values

Given polar coordinates are \(\r = -6\theta = -\frac{\pi}{4}\). Identify the values: r = -6 and \theta = -\frac{\pi}{4}.

Step 2 - Use the conversion formulas

To convert from polar to rectangular coordinates, use the formulas \(x = r \cos(\theta})\) and \(y = r\sin(\theta})\).

Step 3 - Find x coordinate

Substitute the given values into the formula for x: \(x = -6 \cos \left(-\frac{\pi}{4}\right}) = -6 \cdot \frac{\sqrt{2}}{2} = -3\sqrt{2}\).

Step 4 - Find y coordinate

Substitute the given values into the formula for y: \(y = -6 \sin \left(-\frac{\pi}{4}\right}) = -6 \cdot -\frac{\sqrt{2}}{2} = 3\sqrt{2}\).

Step 5 - Write the rectangular coordinates

The rectangular coordinates of the given point are \((-3\sqrt{2}, 3\sqrt{2})\).

Key Concepts

Polar Coordinates

Polar coordinates represent a point in a plane using a distance and an angle. They are written as \( (r, \theta) \), where:

• \( r \) is the radial distance from the origin (the center of the coordinate system) to the point.
• \( \theta \) is the angle measured counterclockwise from the positive x-axis to the line connecting the origin to the point.

For example, the polar coordinates \( (-6, -\frac{\pi}{4}) \) mean the point is 6 units away from the origin and the angle is \( -\frac{\pi}{4} \) or -45 degrees. It’s important to note that negative values for \( r \) place the point in the opposite direction as indicated by the angle.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent a point using horizontal and vertical distances from the origin. They are written as (x, y), where:

• \( x \) is the distance from the y-axis.
• \( y \) is the distance from the x-axis.

For instance, if we have the rectangular coordinates \( (-3\sqrt{2}, 3\sqrt{2}) \), this means the point is \( -3\sqrt{2} \) units to the left of the y-axis and \( 3\sqrt{2} \) units above the x-axis.

Conversion Formulas

To convert polar coordinates to rectangular coordinates, you can use the following formulas:

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

Let's apply these formulas to the example provided:
Start with the given polar coordinates \( (-6, -\frac{\pi}{4}) \).

• The radial distance \( r = -6 \).
• The angle \( \theta = -\frac{\pi}{4} \).

For the x-coordinate:
\( x = -6 \cos(-\frac{\pi}{4}) = -6 \cdot \frac{\sqrt{2}}{2} = -3\sqrt{2} \)
For the y-coordinate:
\( y = -6 \sin(-\frac{\pi}{4}) = -6 \cdot -\frac{\sqrt{2}}{2} = 3\sqrt{2} \)
So, the rectangular coordinates for the polar point \( (-6, -\frac{\pi}{4}) \) are \( (-3\sqrt{2}, 3\sqrt{2}) \).