Problem 62 - Precalculus : Building Concepts and Connections

Question

Convert each of the given polar equations to rectangular form. $$r \cos \theta-3 r \sin \theta=5$$

Step-by-Step Solution

Answer

The rectangular form of the given polar equation \(r \cos \theta - 3r \sin \theta = 5\) is \(x - 3y = 5\)

Step 1: Understanding Polar and Rectangular coordinates

Polar coordinates are represented as (r,\theta), while rectangular coordinates are represented by (x,y). We're given the polar equation \(r \cos\theta-3 r \sin\theta=5\). This needs to be converted into rectangular form. We can do this by substituting the values of x and y, which in polar form are represented by \(r \cos\theta\) and \(r \sin\theta\) respectively.

Step 2: Substitute polar expressions

We replace \(r \cos \theta\) with x and \(r \sin \theta\) with y in the given equation. Our equation then becomes \(x - 3y = 5\)

Step 3: Final Rectangular Equation

And just like that, the polar equation has been converted into rectangular form which is: \(x - 3y = 5\).

Key Concepts

Polar Coordinates

Polar coordinates are a way to express points in a plane using a radius and an angle. They are a fantastic alternative to the more traditional rectangular (or Cartesian) coordinate system. Polar coordinates are denoted as \((r, \theta)\), where:

• \(r\) is the radius, representing the distance from the origin (center point of the coordinate system) to the point in question.
• \(\theta\) (theta) is the angle, measured in radians or degrees, from the positive x-axis to a line connecting the origin to the point.

Think of polar coordinates like a way of describing locations using a map and a compass. If you're standing at the origin, \(r\) tells you how far to walk, and \(\theta\) tells you which direction to head. This system is often used in scenarios involving circular or rotational motion, such as modeling the paths of satellites or the movement of clock hands. Each point can have multiple polar representations since angles can be added/subtracted by full rotations (\(2\pi\) or 360 degrees) without changing the location.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are the most common way of defining points in a plane or space, expressed as \((x, y)\). This system is often preferred for mathematical computations and graphing due to its straightforward nature and ease of understanding.

• \(x\) represents the horizontal distance from the origin.
• \(y\) represents the vertical distance from the origin.

In rectangular coordinates, the entire plane is divided into four quadrants, with axes intersecting at the origin \((0,0)\). Moving right or upwards increases the value of \(x\) or \(y\), while moving left or downwards decreases these values.
Transformation between coordinate systems is a common task in mathematics and is usually done to simplify problems. As shown in the exercise, polar expressions can be converted to rectangular form using the relationships:

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

These equations provide the link between the two systems, allowing for seamless conversion and analysis.

Coordinate Systems

Coordinate systems serve as the foundational structure in geometry, allowing for the precise location of any point on a plane or in space. Having both polar and rectangular coordinate systems gives us flexible tools to tackle mathematical problems from different angles and perspectives.

• Rectangular systems are particularly effective for linear problems where x and y have independent relationships. They suit environments where problems have a direct relationship to these axes, such as plotting graphs of functions.
• Polar systems, on the other hand, are useful for problems involving circles and periodic phenomena. They are best for scenarios where rotation or direction is a significant aspect, such as in navigation or modeling circular motion.

Choosing the appropriate coordinate system can simplify a problem and make computations more straightforward. For any given problem, identifying the predominant geometric feature, like symmetry or periodicity, can guide the decision on which system to use. This strategic choice facilitates an efficient and often more insightful solution approach.