Problem 144 - Calculus Volume 3

Question

Find rectangular coordinates for the given point in polar coordinates. $$ \left(5, \frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

The rectangular coordinates are \((2.5, \frac{5\sqrt{3}}{2})\).

Step 1: Understanding Polar and Rectangular Coordinates

Polar coordinates are expressed as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. To convert to rectangular coordinates \((x, y)\), we use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\).

Step 2: Using Formulas to Calculate x and y

For the given coordinates \((5, \frac{\pi}{3})\), where \(r = 5\) and \(\theta = \frac{\pi}{3}\), we calculate: \(x = 5 \cos(\frac{\pi}{3})\) and \(y = 5 \sin(\frac{\pi}{3})\).

Step 3: Calculating the x-coordinate

We find \(x = 5 \cos(\frac{\pi}{3}) = 5 \times \frac{1}{2} = 2.5\).

Step 4: Calculating the y-coordinate

We find \(y = 5 \sin(\frac{\pi}{3}) = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}\).

Step 5: Writing the Rectangular Coordinates

The rectangular coordinates are \((x, y) = (2.5, \frac{5\sqrt{3}}{2})\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a two-dimensional plane. Rather than using straight-line distances from two axes as in the Cartesian (or rectangular) coordinate system, polar coordinates use a *radius* and an *angle*. The radius, denoted as \( r \), is the distance from a fixed point called the origin. The angle, \( \theta \), is measured from a fixed direction, typically the positive x-axis.This system is particularly useful in scenarios where the relationships in a circular format can simplify calculations or provide more meaningful data than rectangular coordinates.

With polar coordinates \((r, \theta)\):

• \( r \) always represents the non-negative distance from the origin.
• \( \theta \) can be expressed in degrees or radians, usually measured counter-clockwise from the positive x-axis.

Situations such as navigation, where direction and distance are crucial, often use polar coordinates to provide clear, concise information.

Rectangular Coordinates

Rectangular coordinates, better known as Cartesian coordinates in mathematics, write a point by showing how far along the x-axis and y-axis a position is from the origin. These coordinates are seen as \((x, y)\).Each coordinate represents a relative direction:

• \( x \) is the distance along the horizontal axis.
• \( y \) is the distance along the vertical axis.

These coordinates are used extensively because of their intuitive way of describing positions and relationships in two (or more) dimensions.

When converting from polar coordinates to rectangular coordinates:

• Use the formulas: \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \).
• The origin \((0,0)\) in both systems remains the same because it's the reference point.

Understanding rectangular coordinates allows us to seamlessly switch back and forth from various systems, making mathematical expressions and calculations more versatile.

Trigonometric Functions

Trigonometric functions like sine and cosine are essential in mathematics for relating angles to linear distances. They are particularly important in coordinate conversion.When dealing with polar to rectangular conversions, these two functions play a crucial role:

• \( \cos(\theta) \) calculates the adjacent side (x-coordinate) of a right triangle when \( \theta \) is given.
• \( \sin(\theta) \) calculates the opposite side (y-coordinate) of the triangle.

Let's consider the example where \( \theta = \frac{\pi}{3} \), commonly used in problems:

• \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) | Gives the x-coordinate factor.
• \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \) | Determines the y-coordinate factor.

By applying these trigonometric values, converting between polar and Cartesian systems becomes straightforward, and these principles equally extend to more complex problems in fields like physics and engineering.