Problem 48 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \sin \left(\frac{2 \pi}{3}-\theta\right)=5 $$

Step-by-Step Solution

Answer

The Cartesian equation is \(\sqrt{3}x + y = 10\), a line with slope \(-\sqrt{3}\) and y-intercept 10.

Step 1: Understand Polar-Cartesian Relationship

Polar coordinates \((r, \theta)\) can be converted to Cartesian coordinates \((x, y)\) using the formulas \(x = r \cos \theta\) and \(y = r \sin \theta\). The given equation involves \(r\sin(\frac{2\pi}{3} - \theta)\), which hints at transforming using these relationships.

Step 2: Use Trigonometric Identity

The sine of a difference, \(\sin(a - b)\), can be expressed as \(\sin a \cos b - \cos a \sin b\). Applying this to \(r \sin(\frac{2\pi}{3} - \theta)\):\[r \left( \sin\left(\frac{2\pi}{3}\right)\cos\theta - \cos\left(\frac{2\pi}{3}\right)\sin\theta \right) = 5\].

Step 3: Calculate Trigonometric Values

Determine \(\sin\left(\frac{2\pi}{3}\right) = \frac{\sqrt{3}}{2}\) and \(\cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\). Substitute these into the expanded equation: \[r \left( \frac{\sqrt{3}}{2}\cos\theta + \frac{1}{2}\sin\theta \right) = 5\].

Step 4: Convert to Cartesian

Recognize that \(x = r\cos\theta\) and \(y = r\sin\theta\). Replace \(r\cos\theta\) with \(x\) and \(r\sin\theta\) with \(y\): \[\frac{\sqrt{3}}{2}x + \frac{1}{2}y = 5\].

Step 5: Simplify and Identify the Graph

Multiply through by 2 to clear fractions: \[\sqrt{3}x + y = 10\]. This is a linear equation representing a line in the Cartesian plane with a slope of \(-\sqrt{3}\) and a y-intercept at \(10\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to represent points in a two-dimensional plane using a radius and an angle. Instead of describing a point by how far it is along the x and y-axes (as in Cartesian coordinates), polar coordinates determine the location based on a distance from a central point (origin) and an angle from a reference direction, usually the positive x-axis.
In a polar coordinate system, any point is defined by a pair \(r, \theta\), where

• \(r\) is the radial distance from the origin.
• \(\theta\) is the angle measured from the positive x-axis, using either radians or degrees.

The polar coordinates are particularly advantageous when dealing with systems where symmetry and rotations around a point are involved, such as circular or spiral patterns. When the angle \(\theta\) is measured in radians, it makes trigonometric calculations more straightforward.

Cartesian Coordinates

Cartesian coordinates represent points through a pair of numerical coordinates determined by their distances along two perpendicular axes, usually the x and y axes.
This system is often called the rectangular coordinate system. Each point in this coordinate system is described by an ordered pair \(x, y\), where

• \(x\) is the horizontal distance from the origin, along the x-axis.
• \(y\) is the vertical distance from the origin, along the y-axis.

Cartesian coordinates are widely used in various fields like engineering and physics because they easily display linear relationships, which appear as straight lines, making them highly compatible with linear equations.
The transformation from polar to Cartesian therefore often involves using trigonometric functions, as described by the equations \(x = r \cos \theta\) and \(y = r \sin \theta\).

Trigonometric Identities

Trigonometric identities are equations that relate the angles and sides of triangles to fundamental trigonometric functions. These identities are extremely useful in converting equations, such as from polar to Cartesian form, due to their ability to simplify and rearrange expressions.

A key identity used in these transformations is the sine of a difference formula: \( \sin(a - b) = \sin a \cos b - \cos a \sin b\)\.
This identity allows us to express the sine of a difference of two angles in terms of the sines and cosines of the individual angles. In our problem, we used this identity to express \(r \sin(\frac{2\pi}{3} - \theta)\) in terms of \(\cos \theta \) and \(\sin \theta\), which leads to easier conversion into Cartesian coordinates.
The concept is to break down more complex trigonometric expressions into simpler parts that can easily be represented in the Cartesian system.

Linear Equations

Linear equations are mathematical statements that describe straight lines when graphed on a coordinate plane. They can be expressed in the form \(ax + by = c\), where

• \(a\) and \(b\) are coefficients representing the slope's direction.
• \(c\) is a constant term.

A linear equation like \(\sqrt{3}x + y = 10\) represents a straight line. In this equation:

• The slope of the line is \(-\sqrt{3}\).
• The y-intercept, or where the line crosses the y-axis, is at point \(10\).

Linear equations are vital in algebra and geometry as they describe relationships where a change in one variable results in a proportional change in another. They make it easy to predict and analyze trends, relationships, and calculate unknown values. Converting polar equations to this form helps in utilizing straight-line relationships for analysis.