Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ y=6 x $$

Step-by-Step Solution

Answer

The polar equation is \( \tan \theta = 6 \).

Step 1: Understand the Given Equation

We are given the Cartesian equation \( y = 6x \). Our task is to transform this equation, expressed in terms of \( x \) and \( y \), into a polar equation.

Step 2: Recall the Polar-Cartesian Relationships

In polar coordinates, the relationships between polar and Cartesian coordinates are: \( x = r \cos \theta \) and \( y = r \sin \theta \). These will help us convert the Cartesian equation into a polar equation.

Step 3: Substitute Cartesian to Polar

Replace \( y \) with \( r \sin \theta \) and \( x \) with \( r \cos \theta \) in the equation. This gives us: \[ r \sin \theta = 6r \cos \theta \]

Step 4: Simplify the Polar Equation

You can simplify the equation by dividing both sides by \( r \) (assuming \( r eq 0 \)): \[ \sin \theta = 6 \cos \theta \]

Step 5: Express in Standard Polar Form

To convert \( \sin \theta = 6 \cos \theta \) into a standard polar form, divide both sides by \( \cos \theta \): \[ \tan \theta = 6 \] This is the required polar equation.

Key Concepts

Cartesian coordinates

Cartesian coordinates are the building blocks for graphing equations on the standard rectangular coordinate system. This system uses two axes, typically labeled as the x-axis (horizontal) and the y-axis (vertical), to plot points. Each point in this system is defined by an ordered pair,

• The 'x' value indicates the position along the horizontal axis.
• The 'y' value indicates the position along the vertical axis.

Given the equation \( y = 6x \), we see a linear relationship between 'y' and 'x', meaning 'y' is six times the value of 'x'. This represents a straight line passing through the origin with a slope of 6. The slope defines how steep the line is. Remember, in the Cartesian system, equations can easily showcase how 'x' and 'y' interact with each other to form geometric shapes. Breaking it down into smaller steps, as shown, helps reveal these interactions clearly.

conversion to polar coordinates

Converting equations from Cartesian coordinates to polar coordinates can provide a deeper understanding of graphs through different perspectives. Polar coordinates use a different system, expressed as \( (r, \theta) \) where

• 'r' is the radial distance from the origin to the point.
• '\( \theta \)' is the angle measured from the positive x-axis, usually in radians.

To convert the Cartesian equation \( y = 6x \) to a polar equation, we employ the relationships

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

By substituting \( x \) and \( y \) in the equation, it transforms into \( r \sin \theta = 6r \cos \theta \). Simplifying through division by 'r', an assumption based on \( r eq 0 \), leads us to \( \sin \theta = 6 \cos \theta \). This step-by-step conversion is crucial for understanding how Cartesian and polar forms represent the same graph.

polar graph transformation

Transforming Cartesian equations into polar forms often reveals new, insightful patterns and symmetries of the graph. The derived equation, \( \tan \theta = 6 \), represents a straight line cutting through the origin under the polar coordinate system, much like its Cartesian counterpart.

• In polar terms, the line signifies that for every point on this graph, the angle \( \theta \) remains consistent at \( \tan^{-1}(6) \).
• This means no matter how far or close the radial distance 'r' is, the direction (angle) stays constant, emphasizing a unique aspect of polar graphing.

Using polar graph transformation, an entire span of linear relationships found in Cartesian coordinates can be expressed through angle-based symmetry, showcasing the elegance of polar equations in simplifying and visualizing graphs from another dimension. It's a transformation that pivots the focus from distance-based measurements in Cartesian plots to an angle and radiance-centered view in polar graphs.