Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\)-plane. $$ r=\tan \theta $$

Step-by-Step Solution

Answer

The equation is equivalent to \( y = x \) on a Cartesian plane.

Step 1: Convert Polar to Rectangular Coordinates

We start with the polar equation, \( r = \tan \theta \). We know the polar-to-rectangular coordinate conversion formulas are \( x = r \cos \theta \) and \( y = r \sin \theta \). We also have \( \tan \theta = \frac{y}{x} \). Substitute \( r \) and \( \tan \theta \) into these relationships.

Step 2: Substitute for \( r \)

From the given equation, \( r = \tan \theta = \frac{y}{x} \). We can substitute \( r \) as \( \frac{y}{x} \) into the polar conversion equation \( x = r \cos \theta = \frac{y}{x} \cdot \cos \theta \) and \( y = r \sin \theta = \frac{y}{x} \cdot \sin \theta \).

Step 3: Eliminate \( \theta \)

We know \( \tan \theta = \frac{y}{x} \). Therefore, cross-multiply the equation to eliminate \( \theta \): \( y = x\cdot\tan\theta = x\cdot \frac{y}{x} \). Simplify to get \( y = y \).

Step 4: Simplify and Arrive at the Equation

By rearranging and simplifying the equation derived from eliminating \( \theta \), we get the horizontal line equation: \( y = x \), and since we derived a tautology, consider limits (i.e., \( r = \tan\theta \) implies \( r = r \sin\frac{y}{x} \).

Step 5: Graph the Equation in Polar Plane

To sketch the graph, recognize that \( r = \tan \theta \), implies the distance \( r \) between the origin and the graph depends on \( \theta \). It is periodic with period \( \pi \) and exhibits vertical asymptotes at \( \theta = \frac{\pi}{2} + n\pi \), forming a series of radial lines.

Key Concepts

Polar Coordinates

Polar coordinates provide a way to locate points on a plane through two values - the radius \( r \) and the angle \( \theta \). Unlike the rectangular (or Cartesian) system which uses two perpendicular distances, polar coordinates are highly effective for problems involving rotation and angles.
In our specific example, we have the polar equation \( r = \tan \theta \). Here,

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

This approach allows us to graph shapes that rotate around the origin easily. It's especially useful for sketching spirals, circles, and other periodic forms. To graph this equation, the values of \( r \) will change depending on \( \theta \), combining aspects of both distance and direction.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe points using a pair of values \( (x, y) \). Here, \( x \) represents horizontal placement, while \( y \) marks vertical placement.
When converting from polar to rectangular coordinates:

• Use the relation \( x = r \cos \theta \).
• Use the relation \( y = r \sin \theta \).

These formulas help transition from a circular coordinate system to a grid system. For example, for every point in our polar equation \( r = \tan \theta \), using these, we can derive that \( y = x \). This shows that our polar equation corresponds to a line through the origin in rectangular coordinates.

Equation Conversion

Equation conversion, from polar to rectangular or vice versa, is a critical process in mathematics that allows us to switch between different views of a problem, adapting to what fits best.
In the given exercise, we start with \( r = \tan \theta \) in polar format, which shows angle dependency. By converting this into rectangular form, we have \( y = x \).
The key steps for conversion are:

• Identify expressions in terms of \( \theta \) and \( r \).
• Use the relationships \( \tan \theta = \frac{y}{x} \), \( x = r \cos \theta \), and \( y = r \sin \theta \) to replace terms.
• Solve to eliminate \( \theta \) and express the equation solely in terms of \( x \) and \( y \).

By doing this, you can ensure that the geometry being discussed remains intact, yet more comprehensible in a familiar coordinate system.

Graphing Techniques

Graphing techniques are essential tools when interpreting equations, helping visualize the relationship between variables.
For polar equations like \( r = \tan \theta \), a good technique is to:

• Identify the periodic nature from \( \tan \theta \) due to its vertical asymptotes at \( \theta = \frac{\pi}{2} + n\pi \).
• Note that the graph forms a series of radial lines from the origin, showcasing this periodicity.
• Understand the impact of \( r \) - as \( \theta \) changes, so does \( r \), creating a repeated fan-like pattern.

In rectangular coordinates, \( y = x \) simplifies this visualization to a straight line through the origin, confirming how the conversion retains geometric characteristics across coordinate systems.