Problem 27 - Precalculus Mathematics for Calculus

Question

Find the rectangular coordinates for the point whose polar coordinates are given. $$(4, \pi / 6)$$

Step-by-Step Solution

Answer

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

Step 1: Understand Polar Coordinates

Polar coordinates are expressed as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle in radians from the positive x-axis. The given polar coordinates are \((4, \pi/6)\). Here, \(r = 4\) and \(\theta = \pi/6\).

Step 2: Convert Polar to Rectangular Coordinates

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

Step 3: Calculate the x-coordinate

Substitute \(r = 4\) and \(\theta = \pi/6\) into the formula for the x-coordinate: \(x = 4 \cdot \cos(\pi/6)\). Since \(\cos(\pi/6) = \frac{\sqrt{3}}{2}\), the x-coordinate is \(x = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\).

Step 4: Calculate the y-coordinate

Substitute \(r = 4\) and \(\theta = \pi/6\) into the formula for the y-coordinate: \(y = 4 \cdot \sin(\pi/6)\). Since \(\sin(\pi/6) = \frac{1}{2}\), the y-coordinate is \(y = 4 \cdot \frac{1}{2} = 2\).

Step 5: Finalize the Rectangular Coordinates

Combine the calculated x and y values to find the rectangular coordinates: \((x, y) = (2\sqrt{3}, 2)\).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points in a plane using two values:

• The radial distance (\(r\)) from a fixed point called the origin.
• The angular coordinate (\(\theta\)) measured in radians from a fixed direction (usually the positive x-axis).

This system is particularly useful for situations involving circular or rotational symmetries. The given polar coordinate is \((4, \frac{\pi}{6})\).
Here, \(r = 4\) represents the distance from the origin to the point, and \(\theta = \frac{\pi}{6}\) indicates the angle made with the positive x-axis.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe positions in a plane using x and y values.

• The x-coordinate tells us the point’s horizontal distance from the origin.
• The y-coordinate tells us the point’s vertical distance from the origin.

This system is based on a grid layout, or a coordinate plane, which makes it suitable for linear and most practical applications. To convert from polar to rectangular coordinates, you use the following formulas:- \(x = r \cdot \cos(\theta)\)
- \(y = r \cdot \sin(\theta)\)
For the point given as \((4, \frac{\pi}{6})\), the rectangular coordinates are calculated as \((x, y) = (2\sqrt{3}, 2)\). This means that from the origin, the point moves \(2\sqrt{3}\) units along the x-axis and 2 units along the y-axis.

Trigonometric Functions

Trigonometric functions like sine and cosine play a crucial role in the conversion between polar and rectangular coordinates.

• The cosine function, \(\cos(\theta)\), gives the ratio of the adjacent side over the hypotenuse in a right triangle.
• The sine function, \(\sin(\theta)\), gives the ratio of the opposite side over the hypotenuse.

For the polar coordinate \((4, \frac{\pi}{6})\), you will need to calculate:- \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\) This helps us determine the x-coordinate as \(x = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}\).
- \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\)Similarly, this helps us calculate the y-coordinate, \(y = 4 \cdot \frac{1}{2} = 2\).
In general, understanding these functions is key to making accurate conversions and visualizing the relationship between different coordinate systems.