Problem 46 - Precalculus : Building Concepts and Connections

Question

Convert each of the given rectangular equations to polar form. $$3 x+y=1$$

Step-by-Step Solution

Answer

The polar form of the rectangular equation \(3 x+y=1\) is \(r(3 \cos(\theta) + \sin(\theta)) = 1\)

Step 1: Convert Rectangular Coordinates to Polar Coordinates

Start by replacing the x and y terms in the equation with their respective polar coordinate expressions. Remember, \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). The equation becomes \(3r \cos(\theta) + r \sin(\theta) = 1\).

Step 2: Factor Out Common Term r

Next factor out the common term r from the equation in order to isolate it. This yields \(r(3 \cos(\theta) + \sin(\theta)) = 1\)

Step 3: Final Conversion

The final polar form of the rectangular equation \(3 x+y=1\) is \(r(3 \cos(\theta) + \sin(\theta)) = 1\)

Key Concepts

Polar Coordinates

Imagine standing at the center of a circle and pointing to a spot on its edge. This is the essence of polar coordinates: a system that describes a point in a plane using an angle and distance from a fixed center, rather than the more familiar up-and-down, left-and-right movements of rectangular (Cartesian) coordinates.

To understand this, picture a point in the plane. In polar coordinates, we describe this point with two values: the radius (r), which is the straight-line distance from the center (the pole) to the point, and the angle (θ), which is the measure in radians or degrees that you rotate from the positive x-axis (the polar axis) towards the point.

Converting rectangular equations to polar form involves replacing the x and y terms with their polar counterparts: x is replaced with r cos(θ), and y with r sin(θ). This intersection of trigonometry and coordinate systems is a fundamental skill in precalculus and beyond, allowing us to describe and solve problems in different, sometimes more convenient, ways.

Rectangular Coordinates

Intuitively, we use rectangular coordinates when we navigate cities' streets: one number tells us how far east or west we've moved, and the other tells us our movement north or south. Mathematically, these coordinates consist of an ordered pair (x, y), representing horizontal and vertical distances from the origin (0, 0) of a two-dimensional grid.

The power of rectangular coordinates lies in their straightforward representation of points, especially when it comes to plotting shapes and lines or performing algebraic calculations.

However, when dealing with curves, circles, or angles, converting these equations into polar form can drastically simplify calculations. This is because certain equations that produce intricate paths in rectangular coordinates correspond to much simpler forms in polar coordinates, which relates back to the innate circularity of the polar system.

Trigonometry in Precalculus

Trigonometry is the branch of mathematics that deals with triangles, specifically right triangles, and the relationships between their angles and sides. Its concepts are foundational in precalculus, a course designed to prepare students for the rigors of calculus.

Central to trigonometry are the sin (sine), cos (cosine), and tan (tangent) functions, which relate angles to sides of a triangle. These functions are not only crucial for solving geometric problems but also for converting between the polar and rectangular coordinate systems.

For example, in our exercise, we used cos(θ) and sin(θ) to transform the rectangular equation 3x + y = 1 into its polar form. This integration of trigonometry in precalculus paves the way for advanced topics in calculus, physics, and engineering, where polar coordinates often simplify the complexity of certain types of problems.