Question

Write each rectangular equation in polar form. $$3 x-2 y=1$$

Short answer

\(r(3\cos(\theta)-2\sin(\theta))=1\)

Step-by-step solution

Recall Polar Coordinate Relationships

To convert a rectangular equation to polar form, remember the relationships between rectangular coordinates \(x, y\) and polar coordinates \(r, \theta\): \[x = r\cos(\theta)\] and \[y = r\sin(\theta)\]. These will be used to substitute for \(x\) and \(y\) in the given rectangular equation.

Substitute Rectangular Coordinates with Polar Coordinates

Replace the \(x\) and \(y\) in the rectangular equation with the polar coordinate relationships: \[3(r\cos(\theta))-2(r\sin(\theta))=1\].

Simplify the Equation

Factor out the \(r\) from both terms to get the polar form: \[r(3\cos(\theta)-2\sin(\theta))=1\].

Key concepts

Polar Coordinates

Imagine you're at the center of a park and you want to describe the location of a tree. Instead of using directions with a map, consider using a compass and the steps you'd take to walk there: that's the essence of polar coordinates. In this system, every point is defined by how far away, and at what angle, it is from a reference point, called the pole (like your location in the park), and a reference direction, like due north.

Each point is thus given as a pair \(r, \theta\), where \(r\) is the radius or the distance from the pole and \(\theta\) is the angle formed with the reference direction. These coordinates offer an alternative to rectangular (or Cartesian) coordinates, especially in scenarios involving circles or angles, such as plotting the positions of planets around the sun or modeling waves.

Rectangular Coordinates

Rectangular coordinates, the ones that most students learn first, describe the position of a point using two perpendicular lines, typically labeled x (horizontal) and y (vertical). These axes intersect at a point called the origin. Rectangular coordinates are straightforward and intuitive, especially for visualizing points on a grid, solving algebra problems, or when dealing with right angles.

The relationship between a point's x and y coordinates can often be represented by an equation, like \(3x-2y=1\). When we need to switch between systems, such as from rectangular to polar, we use the relationships \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\), which give us a direct link between the linear dimensions of rectangles and the circular dimensions of polar coordinates.

Trigonometry in Polar Form

Trigonometry deals with the relationships between the angles and sides of triangles, and these concepts extend beautifully into polar coordinates. In fact, polar coordinates rely on trigonometric functions to define the relationships between \(r\) and \(\theta\) and the more familiar x and y. By considering the radius as a hypotenuse of a right-angle triangle, we see that \(\cos(\theta)\) and \(\sin(\theta)\) essentially project this radius onto the x and y axes.

When converting equations like \(3x-2y=1\) into polar form, trigonometry becomes our best tool, as we substitute \(x\) and \(y\) with \(r\cos(\theta)\) and \(r\sin(\theta)\). By doing so, we transform rectangular equations into formulas that express circular relationships, revealing how distances and angles play together in a polar framework.