Problem 5 - Thomas Calculus

Question

Sketch the lines in Exercises \(5-8\) and find Cartesian equations for them. $$ r \cos \left(\theta-\frac{\pi}{4}\right)=\sqrt{2} $$

Step-by-Step Solution

Answer

The Cartesian equation of the line is \( x + y = 2 \).

Step 1: Express the given equation in standard polar form

The given equation is \( r \cos \left( \theta - \frac{\pi}{4} \right) = \sqrt{2} \). Use the identity \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \) to expand it. Here, \( \alpha = \theta \) and \( \beta = \frac{\pi}{4} \). Therefore, \( \cos \left( \theta - \frac{\pi}{4} \right) = \cos \theta \cos \frac{\pi}{4} + \sin \theta \sin \frac{\pi}{4} \). Substitute back into the equation: \[ r(\cos \theta \cdot \frac{1}{\sqrt{2}} + \sin \theta \cdot \frac{1}{\sqrt{2}}) = \sqrt{2} \]

Step 2: Simplify the expression

After substitution, simplify the equation: \[ r(\frac{1}{\sqrt{2}} \cos \theta + \frac{1}{\sqrt{2}} \sin \theta) = \sqrt{2} \] Multiply throughout by \( \sqrt{2} \): \[ r(\cos \theta + \sin \theta) = 2 \]

Step 3: Convert to Cartesian coordinates

Recall the polar to Cartesian coordinate transformations: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute them into the equation from Step 2: \[ x + y = 2 \]

Step 4: Sketch the Cartesian line

The equation \( x + y = 2 \) is a line in the Cartesian plane. To sketch it, find two points on the line. When \( x = 0 \), \( y = 2 \). When \( y = 0 \), \( x = 2 \). Plot these points: (0, 2) and (2, 0) and draw the line connecting them. This line has a slope of -1.

Key Concepts

Cartesian Equations

Cartesian equations are a way to represent geometric shapes in the Cartesian coordinate system using algebraic expressions involving x and y.
This is the most common way to describe lines, curves, and shapes on the x-y plane.
In this exercise, we encountered the equation \( x + y = 2 \), which is a simple linear equation.
The Cartesian coordinate system relies on two perpendicular axes:

• The x-axis, which runs horizontally,
• The y-axis, which runs vertically.

Connecting geometric shapes to algebraic equations allows us to use calculations and algebraic manipulation to solve spatial problems.
Additionally, working with Cartesian equations is very useful for graphing these geometric shapes easily on paper or using graphing software.

Trigonometric Identities

Trigonometric identities are vital tools in transforming and simplifying equations, especially when dealing with polar coordinates.
These identities help relate angles and sides of triangles to one another.
In the original exercise, we used the identity for cosine of a difference: \( \cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta \).

• This specific identity allows us to expand expressions involving angles, like \( \theta - \frac{\pi}{4} \), into more manageable parts.
• These expansions can then be substituted back into equations to simplify them or change their form.

Understanding and correctly applying trigonometric identities is crucial in solving problems involving polar coordinates as they are instrumental in bridging between angular presentations and more conventional Cartesian equations.

Polar to Cartesian Conversion

Converting between polar and Cartesian coordinates is an essential skill in mathematics.
It involves transforming a point's representation from one system to another.
For this exercise, we transformed polar coordinates into Cartesian coordinates using two main conversion formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

These formulas link the polar coordinate's radius and angle to the Cartesian coordinate's x and y positions.
By substituting these conversions into the equation obtained after simplification in the polar system, we arrived at a Cartesian equation.
The newly expressed Cartesian equation is what can then be plotted as a line or curve on a plane.

Equation of a Line

The equation of a line is one of the simplest forms in geometry and algebraic representation.
It typically looks like \( y = mx + b \), where m is the slope and b is the y-intercept.
However, lines can also be expressed in other equivalent forms like \( ax + by = c \) or, as it was in this exercise, \( x + y = 2 \).

• The slope-intercept form provides an easy way to see the slope and where the line crosses the y-axis directly.
• The standard form \( ax + by = c \) makes calculations involving intersections and parallel lines more straightforward.

In this particular solution, we found that the line passes through points (0, 2) and (2, 0).
As a result, the slope of the line is -1, indicating a negative inclination as you move from left to right.
Understanding the equation of a line allows one to easily graph and understand the properties of linear relationships.