Problem 27 - Precalculus : Building Concepts and Connections

Question

Convert each of the given pairs of rectangular coordinates to a pair of polar coordinates ( \(r, \theta\) ) with \(r>0\) and \(0 \leq \theta<2 \pi\). $$(1, \sqrt{3})$$

Step-by-Step Solution

Answer

The pair of polar coordinates equivalent to the rectangular coordinates (1, \(\sqrt{3}\)) is (2, 2\(\pi\)/3).

Step 1: Apply the formula for 'r'

First, calculate the radius using the formula \(r = \sqrt{x^2 + y^2}\). In this case, x=1 and y=\(\sqrt{3}\). So, \(r = \sqrt{1^2 + (\sqrt{3})^2} = 2\).

Step 2: Apply the formula for '\(\theta\)'

Next, find the angle using the formula \(\theta = \arctan(y/x)\). However, since \(y > x\) in the given pair of coordinates, in the second quadrant \(\theta = \pi - \arctan(y/x)\). So, \(\theta = \pi - \arctan(\sqrt{3}/1) = \pi - \pi/3 = 2\pi/3\).

Step 3: Write the converted coordinates

The pair of polar coordinates corresponding to the given rectangular coordinates is therefore (r, \(\theta\)) = (2, 2\(\pi\)/3).

Key Concepts

Rectangular Coordinates

Understanding rectangular coordinates is essential when dealing with points represented on a plane using two numbers. These numbers - represented as

• x: The horizontal distance from the origin.
• y: The vertical distance from the origin.

For example, in the pair \((1, \sqrt{3})\), 1 is the x-coordinate indicating the point is 1 unit to the right of the origin, while \(\sqrt{3}\) is the y-coordinate showing the point is \(\sqrt{3}\) units above the origin.
Together, these coordinates define a unique point in the two-dimensional plane. Rectangular coordinates fit perfectly in a Cartesian system, allowing us to use straight grid lines to locate points based on these x and y values.

Trigonometry

Trigonometry is a branch of mathematics focusing on relationships between angles and sides of triangles. It's critical in coordinate conversion.
Particularly, trigonometric functions like sine, cosine, and tangent become significant tools. To convert rectangular coordinates to polar, we use:- The tangent function: \[\theta = \arctan\left(\frac{y}{x}\right)\]This formula helps determine the angle \(\theta\) between the positive x-axis and the line connecting the point to the origin.
To find \(r\), the hypotenuse or radius, we use

• Pythagoras' theorem in a different form: \( r = \sqrt{x^2 + y^2} \).
• This ensures the point's total distance from the origin is accurately calculated regardless of direction.

In our example, these formulas help derive the radius \(r = 2\) and the angle \(\theta = 2\pi/3\).

Coordinate Conversion

Converting coordinates involves changing from rectangular (x, y) to polar coordinates \((r, \theta)\).
The goal is to express the same point either relative to axes or in terms of distance and angle from the origin. This conversion is useful because it provides a different perspective that can simplify analysis or calculations in certain contexts, like physics or engineering.
Here’s how the conversion works:

• Step 1: Determine \(r\) using the formula: \[r = \sqrt{x^2 + y^2}\]
• Step 2: Calculate \(\theta\), the angle: \[\theta = \arctan\left(\frac{y}{x}\right)\] Consider the quadrant if necessary, like in this example where \(\theta = \pi - \arctan(\frac{\sqrt{3}}{1})\).
• Final Step: Express the result as polar coordinates \((r, \theta)\).

This understanding allows you to switch between systems efficiently, making it easier to solve problems across various mathematical applications.