Problem 56 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x-y=3$$

Step-by-Step Solution

Answer

The polar equation is \(r(\cos \theta - \sin \theta) = 3\).

Step 1: Understand the Relationship between Cartesian and Polar Coordinates

Cartesian coordinates are defined as \((x, y)\), where \(x\) and \(y\) are the horizontal and vertical distances from the origin, respectively. Polar coordinates are defined as \((r, \theta)\), where \(r\) is the distance from the origin to the point, and \(\theta\) is the angle from the positive x-axis to the line connecting the origin to the point. The relationships are given by the formulas: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Substitute Polar Coordinates into the Cartesian Equation

Take the given Cartesian equation \(x - y = 3\) and substitute the polar formulas for \(x\) and \(y\): \(r \cos \theta - r \sin \theta = 3\).

Step 3: Simplify the Polar Equation

Factor out \(r\) from the left side of the equation: \(r(\cos \theta - \sin \theta) = 3\). This is the equivalent polar equation.

Key Concepts

Cartesian to Polar Conversion

Converting between Cartesian and polar coordinates is a fundamental skill in various fields of mathematics and physics. Typically in Cartesian coordinates, we describe a point in a two-dimensional plane using

• two values:
  • its horizontal offset from the origin, noted as the x-coordinate, and
  • its vertical offset, which is the y-coordinate.
  This system is intuitive and useful for many practical applications.

In contrast, polar coordinates describe a point by specifying

• the distance from the origin, represented by \( r \), and
• the angle from the positive x-axis, denoted as \( \theta \).

To switch between these systems, you apply the equations:

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

By inserting these polar relationships into a Cartesian equation, you transform it into an equivalent polar form.

Coordinate Systems

Understanding different coordinate systems is vital for graphing and analyzing mathematical equations. The two primary systems are Cartesian and polar.

• **Cartesian System**: This uses perpendicular axes (x and y) to create a grid. Points are identified by their distances from these axes.
• **Polar System**: This system uses a point and an angle. The point's position is determined by how far it is from the origin and its angle relative to the x-axis.

Each system offers unique benefits. The Cartesian system is straightforward when dealing with straight lines and rectangular shapes.
On the other hand, the Polar system excels at describing curves and circles, as it naturally incorporates rotational symmetry.

Equations in Polar Form

Equations written in polar form make it easier to analyze patterns and symmetries of curves. A polar equation relates the radius, \( r \), to the angle \( \theta \), providing a different perspective compared to Cartesian equations.

• **Simplicity**: Polar equations often simplify complex curves. For example, circles and spirals have neat equations in polar form.
• **Applications**: They're useful in fields like engineering and physics where rotational dynamics are common tasks.

To convert from Cartesian to polar form, replace the Cartesian equations with polar relations, such as in the example equation: Given \( x - y = 3 \), converting it to polar form gives \( r (\cos \theta - \sin \theta) = 3 \).
This translation helps in visualizing how lines and curves behave in the polar coordinate system.