Problem 50 - Advanced Engineering Mathematics

Question

Convert the given equation to rectangular cocrdinates. $$ \theta=\pi / 6 $$

Step-by-Step Solution

Answer

The rectangular coordinate equation is \( y = \frac{\sqrt{3}}{3} x \).

Step 1: Recall Definitions

The given polar equation is \( \theta = \frac{\pi}{6} \). In polar coordinates, \( \theta \) represents the angle of a point from the positive x-axis. Our goal is to convert this into rectangular coordinates (x, y).

Step 2: Use Polar to Rectangular Conversion Equations

In polar coordinates, the relationship between polar and rectangular coordinates is given by \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Identify Equation Type

The equation \( \theta = \frac{\pi}{6} \) describes a line through the origin at an angle of \( 30^\circ \) (since \( \frac{\pi}{6} \) radians is equivalent to \( 30^\circ \)) from the positive x-axis.

Step 4: Use the Slope-Intercept Conversion

The line with \( \theta = \frac{\pi}{6} \) can be converted to the slope-intercept form in rectangular coordinates \( y = mx + c \), where \( m \) is the tangent of \( \theta \).

Step 5: Calculate the Slope

Since \( \theta = \frac{\pi}{6} \), the slope \( m \) is \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Step 6: Final Equation in Rectangular Coordinates

Therefore, the equation of the line in rectangular coordinates is \( y = \frac{\sqrt{3}}{3} x \), which is a line through the origin.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are used to pinpoint a location in a plane using two values: an x-coordinate and a y-coordinate. These values tell us how far along and how far up or down the point is from the origin, which is where the x-axis and y-axis cross each other. Think of it as the address of a point in two dimensions.

• The x-coordinate measures horizontal distance from the y-axis.
• The y-coordinate measures vertical distance from the x-axis.

Often represented as $(x, y)$, these coordinates are intuitive for visualizing shapes and motions on a flat surface. They are also handy in algebra for solving equations and plotting graphs.

Polar Coordinates

Polar coordinates provide a different way of locating a point in a plane. Unlike rectangular coordinates, which use x and y values, polar coordinates use an angle and a distance. This system involves two elements: the radial coordinate (r) and the angular coordinate (θ).

• The radial coordinate (r) is the distance from the point to the origin.
• The angular coordinate (θ) represents the angle between the positive x-axis and the line connecting the origin with the point.

This method is particularly beneficial in situations involving circular motions or rotations, where angles offer a natural way to describe positions. Polar plots can simplify problems involving circular or periodic phenomena.

Conversion Equations

Converting between polar coordinates and rectangular coordinates involves specific formulas. These conversion equations help translate one form of representation into another, ensuring different mathematical methods can be used interchangeably. Here are the fundamental conversion formulas:

• To convert from polar to rectangular:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
• To convert from rectangular to polar:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \arctan(\frac{y}{x})\)

These equations allow us to interpret the different ways of describing points in terms of position, motion, or even structure in a given plane.

Trigonometric Identity

Trigonometric identities are mathematical equations that involve trigonometric functions and are true for all values of the occurring variables. In our specific case of converting polar equations like \(\theta = \frac{\pi}{6}\) to rectangular form, the tangent function plays a crucial role.
Recall the identity for tangent:

• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

For example, with \(\theta = \frac{\pi}{6}\), it is converted into rectangular coordinates using the tangent function to determine the slope, as \(\tan \frac{\pi}{6} = \frac{\sqrt{3}}{3}\). Hence, an understanding of these relationships is critical when solving equations that bridge polar and rectangular systems. This understanding ties together these function representations fluidly across different coordinate systems.