Problem 29 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \cos \theta+r \sin \theta=1 $$

Step-by-Step Solution

Answer

The given polar equation is equivalent to the Cartesian line \(x + y = 1\), a straight line graph.

Step 1: Recognize Polar to Cartesian Conversion

In polar coordinates, we can use the conversions: \(x = r \cos \theta\) and \(y = r \sin \theta\). The goal is to replace \(\cos \theta\) and \(\sin \theta\) with \(x\) and \(y\) respectively in the given polar equation.

Step 2: Apply Conversions

Starting with the given equation \(r \cos \theta + r \sin \theta = 1\), apply the substitutions from Step 1:- \(r \cos \theta = x\)- \(r \sin \theta = y\)This transforms the equation into \(x + y = 1\).

Step 3: Identify the Graph

The equation \(x + y = 1\) is in the standard form of a linear equation \(Ax + By = C\), where \(A = 1\), \(B = 1\), and \(C = 1\). This represents a straight line graph cutting the x-axis at \((1, 0)\) and the y-axis at \((0, 1)\).

Step 4: Verification

Verify by substituting back into polar form: If \(x + y = 1\), substituting \(x = r \cos \theta\) and \(y = r \sin \theta\), it gives \(r \cos \theta + r \sin \theta = 1\), which matches the original equation, confirming correctness.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates provide a unique way of describing points in a plane through a combination of a distance from a fixed point and an angle from a fixed direction. Imagine setting out from the origin of a circle, moving a certain distance along a specific angle: that's essentially what polar coordinates allow us to specify.

• The fixed point is often referred to as the pole, similar to the origin in Cartesian coordinates.
• The angle, typically measured in radians or degrees, is taken from a horizontal reference line, usually the positive x-axis.
• For any point in the plane, its position can be described using a radius (r) and an angle (θ).

This method of plotting points is especially useful in scenarios involving circular or rotational symmetry. For example, in problems that reference gadgets or planetary movement, polar coordinates are often the ideal choice.

Cartesian Coordinates

Compared to polar coordinates, Cartesian coordinates might be more familiar. They specify a point in a plane via two perpendicular lines (axes), typically labeled x and y.

• The x-axis traditionally runs left and right (horizontal), while the y-axis runs up and down (vertical).
• Each point is identified uniquely by its distance along the x-axis (horizontal distance) and y-axis (vertical distance) from the origin.
• This coordinate system is known for its simplicity and straightforwardness, making it generally the preferred method for plotting standard points, lines, and shapes.

For instance, in our exercise, when the polar equation was converted to the Cartesian form, it became simply \(x + y = 1\), clearly illustrating a linear equation on the Cartesian plane.

Linear Equation

Linear equations, such as the \(x + y = 1\) from our conversion, are fundamental objects in algebra that represent straight lines when graphed on a Cartesian plane. These equations express a linear relationship between two variables.

• A standard form of a linear equation can be written as \(Ax + By = C\) where A, B, and C are constants.
• The slope-intercept form of \(y = mx + c\) is very common, describing the rate of increase or decrease of x with respect to y.

The simplicity of linear equations makes them incredibly versatile in a host of practical problems, from calculating costs in business to finding distances in physics. In the given exercise, the linear equation reflects the line's intersection points with the axes, creating a clear visual representation.

Graph Identification

Identifying graphs is an essential skill that involves associating an equation with its graphical counterpart. In the context of the exercise, once we converted the polar equation to Cartesian form, recognizing the graph was a crucial step.

• The acknowledgment of the linear equation \(x + y = 1\) as a straight line is fundamental for graph interpretation.
• The line hits the x-axis at (1, 0) and the y-axis at (0, 1), giving it a negative slope, specifically −1, as changes in x yield inversely proportional changes in y.
• This cogent identification helps us not only verify our solution but also enhance our visual understanding of mathematical relationships.

Seeing the alignment with Cartesian coordinates solidifies how equations arise naturally to depict real-world phenomena, bridging the abstract with the tangible.