Problem 33 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \cos \theta+r \sin \theta=1$$

Step-by-Step Solution

Answer

The equivalent Cartesian equation is \( x + y = 1 \). It represents a straight line with a slope of -1 and a y-intercept of 1.

Step 1: Recall Polar to Cartesian Conversion Formulas

To convert from polar to Cartesian coordinates, recall the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). These express the connections between the polar and Cartesian coordinates.

Step 2: Substitute Polar Formulas into Equation

The given polar equation is \( r \cos \theta + r \sin \theta = 1 \). Substitute the conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation. This yields \( x + y = 1 \).

Step 3: Identify the Cartesian Equation

The substitution results in the equation \( x + y = 1 \), which is the equivalent expression in Cartesian coordinates.

Step 4: Describe or Identify the Graph

The equation \( x + y = 1 \) is the equation of a straight line in the Cartesian coordinate system. It has a slope of -1 and a y-intercept of 1. This line passes through the points (1,0) and (0,1).

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing the position of a point in a two-dimensional plane. They use a distance and an angle.

• The 'r' value represents the radial distance from the origin (0,0) to the point.
• The 'θ' (theta) signifies the angle made with the positive x-axis.

In polar coordinates, points are positioned using the format (r, θ).
This system is particularly useful in scenarios involving circular or rotational symmetries.
Some advantages of the polar coordinate system are that it can simplify the mathematics involved in describing curves and regions where other coordinate systems might become cumbersome.

Cartesian Coordinates

Cartesian coordinates rely on x and y values to specify the location of a point.

• The x-coordinate indicates the horizontal distance from the origin.
• The y-coordinate indicates the vertical distance from the origin.

Points are noted in the format (x, y) and can easily describe the geometry of shapes and lines.
Cartesian coordinates are widely used in algebra and calculus because they provide a straightforward method for plotting points and visualizing algebraic equations.Understanding the conversion from polar to Cartesian coordinates involves simple substitutions: \( x = r \cos \theta \) and \( y = r \sin \theta \). These substitutions allow for translating figures and solving problems in different coordinate systems, broadening the mathematical analysis.

Equation of a Line

The equation \( x + y = 1 \) represents a straight line in the Cartesian coordinate system.

• This equation is in the standard form of a linear equation, \( ax + by = c \).
• In this form, the slope \( m \) of the line is \(-\frac{a}{b} \), which for \( x + y = 1 \) is -1.
• The line intercepts the y-axis at the point (0,1), showcasing where the line crosses this axis.

Since it covers both axes by crossing them at two strategic points, the line provides insight into questions of symmetry and translation properties in Cartesian coordinates. Finding the line's slope and intercept are crucial steps in understanding how a line behaves across the x-y plane.

Graph Identification

Identifying graphs involves recognizing the forms and features in Cartesian coordinates to visualize the mathematical representation.
Given the equation \( x + y = 1 \), you can identify its graph as a line. It is known for its simplicity in linear graphs.

• The slope is -1, indicating a downward diagonal direction.
• The y-intercept is at (0,1), making it easy to plot the starting point of the line on the graph.

To draw the line, plot the intercepts and connect them with a straight edge. Recognizing these elements on a graph can help clarify the relationship between algebraic equations and their geometric interpretations.
This identification is crucial for solving and understanding problems in both academic settings and real-world applications where data visualization plays a key role.