Problem 61 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$2 r \cos \theta+r \sin \theta=4$$

Step-by-Step Solution

Answer

The Cartesian (rectangular) form of the given polar equation \(2 r \cos(\theta) + r \sin(\theta) = 4\) is \(2x + y = 4\).

Step 1: Recognize and Use the Mathematical Relations Between Polar and Rectangular Coordinates

Recall the relations between the polar coordinates \((r, \theta)\) and the rectangular coordinates \((x, y)\), which are \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). These will be used in the next step to transform the polar equation into a rectangular one.

Step 2: Substitute Polar Coordinate Values in the Provided Equation

Substitute the values of \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\) into the given equation \(2 r \cos(\theta) + r \sin(\theta) = 4\). When substituted, we get \(2x + y = 4\).

Step 3: Rearrange the Equation to a Recognizable Rectangular Form

The equation obtained in Step 2 is already in a standard rectangular form. So no further modification is required. The rectangular form of the given polar equation is thus \(2x + y = 4\).

Key Concepts

Polar Coordinates

Polar coordinates describe a point in a plane using a distance and an angle.
Imagine a point on a piece of paper. Instead of describing where it is with a typical x and y coordinate, we use:

• Distance: How far the point is from the origin (0,0).
• Angle: The direction of the point relative to a reference direction, typically the positive x-axis.

The coordinates are written as \((r, \theta)\), where:

• r: Represents the radial distance from the origin to the point.
• \(\theta\): Denotes the angle measured in radians from the positive x-axis.

This system is particularly useful in scenarios involving circular motion or symmetry around a central point, making calculations more intuitive for certain problems.

Rectangular Coordinates

Rectangular coordinates offer a different way to describe a point in the plane using two perpendicular axes.
Think about plotting a graph using an x and y grid. This familiar system tells us:

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

These coordinates are expressed as \((x, y)\). This system is very handy for straightforward calculation and is frequently used in algebra and geometry.
It helps simplify problems that involve straight lines or direct linear relationships, as it relies on a Cartesian coordinate system that is universally recognized.

Coordinate Transformation

Coordinate transformation allows us to switch between different coordinate systems, like polar and rectangular.
This process involves using mathematical relationships:

• From Polar to Rectangular: To convert polar coordinates \((r, \theta)\) to rectangular coordinates \((x, y)\), use:
  • \(x = r \cos(\theta)\)
  • \(y = r \sin(\theta)\)
• From Rectangular to Polar: To convert rectangular coordinates \((x, y)\) back to polar, use:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

These transformations are crucial in solving many mathematical and engineering problems as they enable us to view and solve geometrical problems from multiple perspectives.
By understanding both systems, you can tackle problems that look complicated in one system but simple in another.