Problem 33 - Precalculus Mathematics for Calculus

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

The given point in polar coordinates is \((6 \sqrt{2}, 11\pi/6)\). The first number, \(6 \sqrt{2}\), represents the radius \(r\), and the second number, \(11\pi/6\), represents the angle \(\theta\), measured counterclockwise from the positive x-axis.

Step 2: Determine Formulas for Conversion

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

Step 3: Calculate the x-coordinate

Substitute \(r = 6 \sqrt{2}\) and \(\theta = 11\pi/6\) into the formula for \(x\):\[x = 6 \sqrt{2} \cos(11\pi/6)\]Using the cosine value for \(11\pi/6\), which is \(\cos(11\pi/6) = \sqrt{3}/2\), compute:\[x = 6 \sqrt{2} \times \frac{\sqrt{3}}{2} = 3\sqrt{6}\]

Step 4: Calculate the y-coordinate

Substitute \(r = 6 \sqrt{2}\) and \(\theta = 11\pi/6\) into the formula for \(y\):\[y = 6 \sqrt{2} \sin(11\pi/6)\]Using the sine value for \(11\pi/6\), which is \(\sin(11\pi/6) = -1/2\), compute:\[y = 6 \sqrt{2} \times \left( -\frac{1}{2} \right) = -3\sqrt{2}\]

Step 5: Write the Rectangular Coordinates

After calculating the x and y coordinates, we find the rectangular coordinates:\((x, y) = (3\sqrt{6}, -3\sqrt{2})\).

Key Concepts

Understanding Polar Coordinates

Polar coordinates are a way of representing a point in a plane using a distance from a fixed point and an angle from a fixed direction. The fixed point is known as the pole, often corresponding to the origin in rectangular coordinates, and the fixed direction is usually the positive x-axis.

In polar coordinates, a point is represented by

• the radius \(r\), which is the distance from the pole, and
• the angle \(\theta\), which is measured in radians counterclockwise from the positive x-axis.

For example, a point expressed as \((6 \sqrt{2}, 11\pi/6)\) means that it is \(6 \sqrt{2}\) units away from the origin, and the angle formed with the positive x-axis is \(11\pi/6\) radians.

Understanding polar coordinates is essential for converting them into rectangular coordinates, where we use standard pairs to define locations on a plane.

Converting to Rectangular Coordinates

Rectangular coordinates use the familiar Cartesian system, where points are represented using an x-coordinate (horizontal position) and a y-coordinate (vertical position). The conversion from polar to rectangular coordinates is based on trigonometric functions, specifically the sine and cosine functions.

To switch from polar \((r, \theta)\) to rectangular \((x, y)\), we use the formulas:

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

In the given example, the radius \(r\) is \(6 \sqrt{2}\) and the angle \(\theta\) is \(11\pi/6\). By substituting into these formulas, we calculate:

• \(x = 6 \sqrt{2} \cdot \cos(11\pi/6) = 3\sqrt{6}\)
• \(y = 6 \sqrt{2} \cdot \sin(11\pi/6) = -3\sqrt{2}\)

These give us the rectangular coordinates \((3\sqrt{6}, -3\sqrt{2})\), a common way to express points in regular geometry.

Role of Trigonometric Functions in Conversion

Trigonometric functions, such as sine and cosine, are crucial for converting coordinates because they relate the angles in polar coordinates to the x and y lengths in rectangular coordinates.

The cosine function \(\cos \theta\) gives the horizontal distance (or projection along the x-axis), while the sine function \(\sin \theta\) gives the vertical distance (or projection along the y-axis).

In our example of \(\theta = 11\pi/6\), we find that:

• The cosine \(\cos(11\pi/6)\) equals \(\sqrt{3}/2\). This value is used to compute the x-coordinate.
• The sine \(\sin(11\pi/6)\) equals \(-1/2\). This helps in determining the y-coordinate.

These calculations illustrate how trigonometric functions allow seamless transformation from one system of coordinates to another, enabling a complete understanding of the position of a point in a plane.