Problem 80 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ \theta=\frac{2 \pi}{3} $$

Step-by-Step Solution

Answer

The polar equation \( \theta = \frac{2\pi}{3} \) converts to the rectangular line equation \( x = -\sqrt{3}y \).

Step 1: Understand Polar Coordinates

In polar coordinates, a point is represented by \[ (r, \theta) \] where \( r \) is the distance from the origin and \( \theta \) is the angle. The task is to convert the given polar equation into rectangular coordinates (\( x, y \)). We have \( \theta = \frac{2\pi}{3} \) without a specific \( r \) value, which means we have an angle direction but no specific radius.

Step 2: Use Relationship Between Polar and Rectangular Coordinates

The relationships between polar and rectangular coordinates are given by:\[ x = r \cos(\theta) \]\[ y = r \sin(\theta) \]where \( x \) and \( y \) are the rectangular coordinates. Since no \( r \) is specified, focus on expressing \( \cos(\theta) \) and \( \sin(\theta) \) to find the line corresponding to \( \theta = \frac{2\pi}{3} \).

Step 3: Calculate Trigonometric Values

First, find \( \cos(\theta) \) and \( \sin(\theta) \) for \( \theta = \frac{2\pi}{3} \). These are fundamental trigonometric positions:\[ \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \]\[ \sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2} \]

Step 4: Rewrite Equation in Rectangular Form

From the known relationships:- \( x = r \cos(\theta) \) becomes \( -x/2 \) when in terms of the given angle.- \( y = r \sin(\theta) \) becomes \( y\sqrt{3}/2 \).Since there’s no given \( r \), these variables relate as \[ y\sqrt{3} = -x \]This can be simplified as\[ \sqrt{3}y = -x \] or \[ x = -\sqrt{3}y \]representing the line through the origin with slope \(-\frac{1}{\sqrt{3}}\).

Step 5: Conclusion: Rectangular Equation

The equation \( \theta = \frac{2\pi}{3} \) in polar coordinates is a line with direction angle only (angle without specific radius). It converts to the rectangular form: \[ x = -\sqrt{3}y \]

Key Concepts

Polar Coordinates

Polar coordinates are a way to locate a point in the plane using a distance and an angle. Instead of the familiar

• Cartesian system with axes and coordinates
• polar coordinates are given as a pair
• \((r, \theta)\) where:
  • \(r\) represents the radial distance from a fixed point called the origin.
  • \(\theta\) denotes the angle from a fixed direction, usually measured in radians from the positive x-axis.

This system is particularly useful for situations where length and angle measurement are more relevant than direct coordinates.
For example, when dealing with circular motion or spiral shapes.
In the exercise at hand, we have a given angle, \(\theta = \frac{2\pi}{3}\), but no specific radius.This implies we are focusing purely on the direction — representing a line rather than a specific point.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are defined by a pair

• \((x, y)\) that describe a point's exact position on a plane using horizontal and vertical axes:
  • \(x\) is the horizontal distance from the origin.
  • \(y\) is the vertical distance.

These coordinates are straightforward, making them a standard choice for many mathematical and engineering problems.
They allow you to easily calculate distances and angles using algebra and geometry.
The exercise aims to transition from a polar representation where an angle is given, to this format. By doing so, we pinpoint how a direction defined in polar terms transforms into a line equation in rectangular terms.

Trigonometric Functions

Trigonometric functions are key to translating between polar and rectangular coordinates.
They relate the angles in polar coordinates to the respective x and y positions in rectangular coordinates.

• \(\cos(\theta)\) helps determine the horizontal projection, and \(\sin(\theta)\) the vertical.

In our exercise, we're given \(\theta = \frac{2\pi}{3}\).
Calculating these gives:

• \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\), indicating an acute negative horizontal component.
• \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\), indicating a positive vertical component.

These values direct how we rewrite our direction as a straight line, bringing the angle into a rectangular context.

Coordinate Transformation

Coordinate transformation involves changing the description of a location from one coordinate system to another.
For problems like this exercise, it specifically addresses converting a point or line from polar to rectangular notation.

• The process utilizes relationships:
  • \(x = r \cos(\theta)\)
  • \(y = r \sin(\theta)\)

Since \(r\) is not given in the original exercise, the interest lies in understanding directional components.
These relations transform into equations that describe the tilt of the line defined by the angle \(\theta = \frac{2\pi}{3}\).
This slide in representation gives equation \(x = -\sqrt{3}y\), a mathematical shift that helps visualize the directional nature of the original polar statement.