Problem 26 - Precalculus: Functions and Graphs

Question

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 \( \theta = \arctan(6) \).

Step 1: Understand the Rectangular Equation

The given equation is in rectangular form as \( y = 6x \). This is a linear equation that represents a line with a slope of 6 passing through the origin.

Step 2: Recall Conversion Formulas

In polar coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \). These relationships are used to convert rectangular equations to polar form.

Step 3: Substitute Using Conversion Formulas

Replace \( x \) with \( r \cos \theta \) and \( y \) with \( r \sin \theta \) to rewrite the equation as: \[ r \sin \theta = 6 (r \cos \theta) \]

Step 4: Simplify the Polar Equation

Factor out \( r \) (assuming \( r eq 0 \)) to obtain: \[ \sin \theta = 6 \cos \theta \]Divide both sides by \( \cos \theta \) to get: \[ \tan \theta = 6 \]

Step 5: Final Polar Equation Form

The polar equation that corresponds to \( y = 6x \) is \( \theta = \arctan(6) \), which represents a line passing through the origin with an angle of \( \arctan(6) \) from the positive x-axis.

Key Concepts

Rectangular to Polar Conversion

In mathematics and precalculus, converting between rectangular and polar coordinates is a valuable tool. Rectangular coordinates use the familiar

• \( x \)
• \( y \)

axes system and describe a point's position in a two-dimensional plane in terms of its horizontal and vertical distances from the origin.
Conversely, polar coordinates describe a point’s position in terms of

• \( r \) (the distance from the origin)
• \( \theta \) (the angle from the positive x-axis)

To convert a point from rectangular to polar form, use these relationships:

• \( x = r\cos\theta \)
• \( y = r\sin\theta \)
Additionally, the distance formula
• \( r = \sqrt{x^2 + y^2} \)
and angle formula
• \( \theta = \arctan\left(\frac{y}{x}\right)\)
are crucial for finding polar coordinates from a given rectangular point.
The example given switches the linear equation \( y = 6x \) from a rectangular to a polar form by replacing \( x \) and \( y \) with their polar equivalents: \( r \cos \theta \) and \( r \sin \theta \).

Linear Equations in Polar Form

Linear equations can be expressed in polar form, which is particularly useful for visualizing their orientation and slope with respect to the origin. Here, we started with a simple linear equation in rectangular form: \( y = 6x \).
To convert it into polar form:

• We substitute \( x \) by \( r\cos(\theta) \)
• Substitute \( y \) by \( r\sin(\theta) \)

This leads to the equation \( r\sin(\theta) = 6r\cos(\theta) \). By factoring out \( r \), assuming \( r eq 0 \), we simplify it to \( \tan(\theta) = 6 \).
Ultimately, in polar form, the equation is expressed as \( \theta = \arctan(6) \). This indicates a line through the origin at an angle of \( \arctan(6) \) with the x-axis, effectively showcasing the direction and steepness of this line in the polar coordinate system.

Trigonometric Functions

Trigonometric functions play a critical role when dealing with polar coordinates and their conversion from rectangular coordinates. The three primary functions are:

• \( \sin \)
• \( \cos \)
• \( \tan \)

They relate angles and distances in polar coordinates to the x and y coordinates in rectangular systems.
The function \( \tan \theta = \frac{y}{x} \) is particularly helpful when converting linear equations, enabling us to extract angles based on the ratio of y to x coordinates.
In our problem, it helped simplify the expression \( \frac{\sin \theta}{\cos \theta} = 6 \) to simply \( \tan \theta = 6 \), giving a straightforward interpretation of the line's angle. Understanding how these trigonometric functions interact not only helps in conversions but also in deepening the overall comprehension of trigonometry in analytical geometry.

Conversion Formulas in Precalculus

In precalculus courses, conversion formulas between polar and rectangular coordinates are fundamental. These formulas allow students to translate between the two systems, each useful in different contexts. The key conversion formulas used include:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan \left( \frac{y}{x} \right) \)

These conversions can turn complex problems involving angles and distances into simpler form, perfect for graphing or solving equations.
For instance, with the linear equation \( y = 6x \), using these formulas helps reveal its properties in polar form as \( \theta = \arctan(6) \). These abilities not only benefit math skills but also physics and engineering tasks where varying coordinate systems are used.