Problem 42 - Precalculus with Calculus Previews

Question

Find a polar equation that has the same graph as the given rectangular equation. $$ 3 x+8 y+6=0 $$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{-6}{8\sin(\theta) + 3\cos(\theta)}\).

Step 1: Understand the Rectangular Equation

The given rectangular equation is \(3x + 8y + 6 = 0\). This is a linear equation representing a line in the Cartesian (rectangular) coordinate system.

Step 2: Express y as a Function of x

Rearrange the equation to solve for \(y\). Subtract \(3x + 6\) from both sides to get \(8y = -3x - 6\). Then, divide every term by 8 to isolate \(y\), resulting in \(y = -\frac{3}{8}x - \frac{3}{4}\).

Step 3: Convert to Polar Coordinates

Recall the conversions: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Substitute these into the expression for \(y\): \(r\sin(\theta) = -\frac{3}{8}(r\cos(\theta)) - \frac{3}{4}\).

Step 4: Simplify the Polar Equation

Multiply the entire equation by 8 to eliminate fractions: \(8r\sin(\theta) = -3r\cos(\theta) - 6\). Rearrange to group \(r\) terms: \(8r\sin(\theta) + 3r\cos(\theta) = -6\).

Step 5: Factor Out r

Notice that \(r\) can be factored out from the left side of the equation: \(r(8\sin(\theta) + 3\cos(\theta)) = -6\).

Step 6: Solve for r

Divide both sides of the equation by \(8\sin(\theta) + 3\cos(\theta)\) to isolate \(r\): \(r = \frac{-6}{8\sin(\theta) + 3\cos(\theta)}\).

Key Concepts

Rectangular to Polar Conversion

Rectangular to Polar conversion is an essential process for transforming equations from the rectangular coordinate system to the polar coordinate system. The rectangular coordinate system, also known as the Cartesian coordinate system, uses an

• x-axis for horizontal positioning, and
• y-axis for vertical positioning.

In contrast, the polar coordinate system uses:

• \( r \), which is the distance from the origin to the point, and
• \( \theta \), which is the angle measured from the positive x-axis.

To convert from rectangular coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the equations:

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

By using these conversions, you can transform any rectangular equation into a polar equation, which is often useful in scenarios involving rotational symmetry. This transformation is straightforward once you understand the geometrical basis, making it easier to portray conic sections, spirals, and other plots in polar form.

Linear Equations in Polar Coordinates

Linear equations can also be expressed in polar coordinates, although they appear more complex than their rectangular counterparts. When a linear equation in the rectangular system, like \( 3x + 8y + 6 = 0 \), is converted into polar form, it still represents a straight line, but its expression will involve trigonometric functions.
The conversion process substitutes \( x \) and \( y \) with their polar equivalents, \( r\cos(\theta) \) and \( r\sin(\theta) \) respectively. This leads to equations combining both \( r \) and \( \theta \), such as in the form:

• \( r(8\sin(\theta) + 3\cos(\theta)) = -6 \)
• \( r= \frac{-6}{8\sin(\theta) + 3\cos(\theta)} \)

Seeing linear equations in polar form can initially appear daunting, but these expressions succinctly encapsulate the geometry of lines in a system based upon angles and distances. They are particularly useful in settings like navigation and mechanical engineering, where directions relative to a point are pivotal.

Polar Coordinate System

The Polar coordinate system is a two-dimensional coordinate system where each point on a plane is determined by its distance from a reference point, known as the pole, and an angle relative to a reference direction. Unlike the rectangular coordinate system, which can seem direct with its perpendicular axes and grid layout, the polar system is very useful for problems involving circular or rotational motion.
In the polar system:

• \( r \) represents the radial distance from the pole to the point, and
• \( \theta \) signifies the counterclockwise angle from the positive x-axis.

This system is advantageous in physics and engineering for describing phenomena related to rotation. It is often more intuitive to describe an object's position or path in polar coordinates when dealing scenarios like circular paths or motion around a central point. Understanding and being able to switch to the polar coordinate system expands both a student's toolkit and their understanding of spatial geometry.