Problem 44 - College Trigonometry

Question

Convert the point from rectangular coordinates into polar coordinates with \(r \geq 0\) and \(0 \leq \theta<2 \pi\). $$ \left(\frac{\sqrt{3}}{4},-\frac{1}{4}\right) $$

Step-by-Step Solution

Answer

The polar coordinates are \(\left(\frac{1}{2}, \frac{11\pi}{6}\right)\)."

Step 1: Understanding Rectangular to Polar Conversion

Polar coordinates are in the form \((r, \theta)\), where \(r\) is the radial distance from the origin and \(\theta\) is the angle measured from the positive x-axis. Our task is to find \(r\) and \(\theta\) from the given rectangular coordinates \(\left(x, y\right) = \left(\frac{\sqrt{3}}{4}, -\frac{1}{4}\right)\).

Step 2: Calculate the Radial Distance \(r\)

The formula for \(r\) in polar coordinates is \(r = \sqrt{x^2 + y^2}\). Substitute \(x = \frac{\sqrt{3}}{4}\) and \(y = -\frac{1}{4}\) into the formula:\[r = \sqrt{\left(\frac{\sqrt{3}}{4}\right)^2 + \left(-\frac{1}{4}\right)^2} = \sqrt{\frac{3}{16} + \frac{1}{16}} = \sqrt{\frac{4}{16}} = \frac{1}{2}\.\]

Step 3: Calculate the Angle \(\theta\)

The angle \(\theta\) is found using \(\tan \theta = \frac{y}{x}\). Substitute \(x = \frac{\sqrt{3}}{4}\) and \(y = -\frac{1}{4}\):\[\tan \theta = \frac{-\frac{1}{4}}{\frac{\sqrt{3}}{4}} = -\frac{1}{\sqrt{3}}\.\]The reference angle \(\alpha\) where \(\tan \alpha = \frac{1}{\sqrt{3}}\) is \(\frac{\pi}{6}\). However, since the y-value is negative, \(\theta\) should be in the fourth quadrant, hence \(\theta = -\frac{\pi}{6}\). Adding \(2\pi\) to make it positive, \(\theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}\).

Step 4: Combine \(r\) and \(\theta\) into Polar Coordinates

Now that we have both \(r = \frac{1}{2}\) and \(\theta = \frac{11\pi}{6}\), we can express the polar coordinates as \(\left(\frac{1}{2}, \frac{11\pi}{6}\right)\).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to locate a point in a plane using an ordered pair \( (x, y) \). Each point on the plane can be represented by these coordinates based on two perpendicular lines called axes: the x-axis (horizontal) and the y-axis (vertical).
The x-coordinate represents the horizontal distance from the origin, where the two axes intersect. The y-coordinate indicates the vertical distance from the origin.
In our exercise, the rectangular coordinates of the point are given as \( \left(\frac{\sqrt{3}}{4}, -\frac{1}{4}\right) \). This means that the point is \( \frac{\sqrt{3}}{4} \) units right of the origin and \( \frac{1}{4} \) units below the origin.

Radial Distance

In polar coordinates, the radial distance, often represented as \( r \), is the distance from the origin to the point. This is a key component of the point's location in polar coordinates. Unlike rectangular coordinates, where distance is divided into horizontal and vertical components, the radial distance is a direct measure from the origin to the point.

To calculate \( r \), we use the formula for distance in terms of \( x \) and \( y \): \[ r = \sqrt{x^2 + y^2} \].
Using the given coordinates \( (x, y) = \left(\frac{\sqrt{3}}{4}, -\frac{1}{4}\right) \):

• Compute the square of each coordinate: \( x^2 = \left(\frac{\sqrt{3}}{4}\right)^2 = \frac{3}{16} \)
• Compute \( y^2 = \left(-\frac{1}{4}\right)^2 = \frac{1}{16} \)
• Add these results: \( \frac{3}{16} + \frac{1}{16} = \frac{4}{16} = \frac{1}{4}\)
• Take the square root: \( r = \sqrt{\frac{1}{4}} = \frac{1}{2} \)

Thus, the radial distance \( r \) is \( \frac{1}{2} \).

Angle Calculation

The angle in polar coordinates, denoted as \( \theta \), represents the direction from the origin to the point. It’s measured from the positive x-axis, moving counterclockwise.

To find \( \theta \), we use the tangent function: \[ \tan \theta = \frac{y}{x} \].
Substituting the given values, \( x = \frac{\sqrt{3}}{4} \) and \( y = -\frac{1}{4} \):

• Calculate \( \tan \theta = \frac{-\frac{1}{4}}{\frac{\sqrt{3}}{4}} = -\frac{1}{\sqrt{3}} \)
• The reference angle \( \alpha \) satisfies \( \tan \alpha = \frac{1}{\sqrt{3}} \), giving \( \alpha = \frac{\pi}{6} \)

However, because the y-value is negative, the point is in the fourth quadrant, where angles are measured in the clockwise direction. So, \( \theta = -\frac{\pi}{6} \). To keep \( \theta \) within the standard range \( [0, 2\pi) \), add \( 2\pi \), giving us \( \theta = \frac{11\pi}{6} \).

Coordinate Conversion

Converting between rectangular and polar coordinates allows us to describe a point in two different ways based on the context.
For a given point \( (x, y) \), to convert to polar coordinates \( (r, \theta) \), follow these steps:

• First, calculate the radial distance using the formula \( r = \sqrt{x^2 + y^2} \).
• Then, determine the angle using \( \theta = \arctan \left(\frac{y}{x}\right) \). Adjust \( \theta \) based on the quadrant in which the point lies.

This process transforms Cartesian descriptions into polar ones, useful in circumstances involving rotation or cyclical paths, like circles. In our exercise, the rectangular coordinates \( \left(\frac{\sqrt{3}}{4}, -\frac{1}{4}\right) \) were converted into polar coordinates \( \left(\frac{1}{2}, \frac{11\pi}{6}\right) \), providing a concise view of the distance from the origin and angular direction.