Problem 31 - Algebra and Trigonometry

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (6 \sqrt{2}, 11 \pi / 6) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((3\sqrt{6}, -3\sqrt{2})\).

Step 1: Understanding Polar Coordinates

Polar coordinates are of the form \((r, \theta)\), where \(r\) is the radius and \(\theta\) is the angle measured from the positive x-axis. Here, \(r = 6\sqrt{2}\) and \(\theta = \frac{11\pi}{6}\). The task is to convert these polar coordinates to rectangular coordinates which are expressed in the form \((x, y)\).

Step 2: Formulas for Conversion

To convert polar coordinates to rectangular coordinates, use the formulas: \[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]These equations help in calculating the x and y values using the given \(r\) and \(\theta\).

Step 3: Calculate the x-coordinate

Substitute the given values into the formula for the x-coordinate:\[ x = r \cos(\theta) = 6\sqrt{2} \cos\left(\frac{11\pi}{6}\right) \]Knowing that \(\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}\) (since it corresponds to a 30° reference angle or \(\frac{\pi}{6}\), and cos is positive in the fourth quadrant), we compute:\[ x = 6\sqrt{2} \times \frac{\sqrt{3}}{2} = 3\sqrt{6} \]

Step 4: Calculate the y-coordinate

Substitute the given values into the formula for the y-coordinate:\[ y = r \sin(\theta) = 6\sqrt{2} \sin\left(\frac{11\pi}{6}\right) \]Since \(\sin\left(\frac{11\pi}{6}\right) = -\frac{1}{2}\) (sin is negative in the fourth quadrant at this angle), we compute:\[ y = 6\sqrt{2} \times -\frac{1}{2} = -3\sqrt{2} \]

Step 5: Final Rectangular Coordinates

Thus, the rectangular coordinates of the point are:\((x, y) = (3\sqrt{6}, -3\sqrt{2})\).

Key Concepts

Polar Coordinates

Polar coordinates are a unique way to describe a point's position in the plane using a distance and an angle. Imagine standing at the origin of a map (like the North Pole), that's your reference point. The distance from this point to where you are is called the radius, denoted as \(r\). The angle \(\theta\) is how much you need to "turn" from the positive x-axis to get to the line where this distance is measured. Think of \(\theta\) as the direction a compass is pointing. These coordinates are written as \((r, \theta)\).
For example, given \((6\sqrt{2}, 11\pi / 6)\), it means you move 6 times the length of \(\sqrt{2}\) units away from the origin, and turn at an angle of \(\frac{11\pi}{6}\) radians from the positive x-axis. It's like polar exploration on a flat surface!

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, are much like points in a regular grid. They describe positions using horizontal and vertical lines, typical of graph paper. Rectangular coordinates are written as \((x, y)\), showing how far right (positive x) or left (negative x) and how far up (positive y) or down (negative y) a point is from the origin.
Think of these coordinates as your typical street map, where an address indicates how many blocks to walk east/west and north/south. Our goal often involves converting from polar to rectangular, finding \(x\) and \(y\) using math that relates distance and angles to straightforward grid positions.

Trigonometric Functions

Trigonometric functions are essential tools in converting coordinates because they relate angles and distances. The functions \(\cos\) and \(\sin\) help translate the polar coordinates' radius and angle into rectangular coordinates' x and y. If you're spinning that compass, \(\cos(\theta)\) tells how far east or west (x-axis) you move, and \(\sin(\theta)\) tells how far north or south (y-axis) you go.
So in the formula \(x = r \cos \theta\), \(\cos(\theta)\) helps us draw that east-west movement on our grid, and for \(y = r \sin \theta\), \(\sin(\theta)\) gives us that north-south line. Memorizing the unit circle values, like knowing \(\cos\left(\frac{11\pi}{6}\right) = \frac{\sqrt{3}}{2}\), gives quick solutions for the conversion task!

Coordinate System Conversion

Converting coordinates between different systems, such as from polar to rectangular, is crucial for understanding different perspectives in math and physics. This conversion involves using trigonometric functions to translate a point's angular and radial position into x and y positions.
The key formulas to remember:

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

By plugging in the values of \(r\) and \(\theta\), you get the rectangular coordinates. For example, with \(r = 6\sqrt{2}\) and \(\theta = \frac{11\pi}{6}\), you calculate \(x = 3\sqrt{6}\) and \(y = -3\sqrt{2}\). This method seamlessly transforms polar data into a format that's often more practical for plotting on a typical x-y plane.