Problem 16 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x=5$$

Step-by-Step Solution

Answer

The polar equation is \(r\cos\theta = 5\).

Step 1: Identify Cartesian coordinates

The given equation is in Cartesian coordinates: \(x = 5\). This represents a vertical line at \(x = 5\) in the Cartesian plane.

Step 2: Recall Polar coordinates relationship

In polar coordinates, the relationships with Cartesian coordinates are \(x = r\cos\theta\) and \(y = r\sin\theta\). We need to convert the Cartesian equation to polar form.

Step 3: Substitute into polar form

Substitute \(x = r\cos\theta\) into the given equation \(x = 5\). This yields the equation: \[r\cos\theta = 5\]

Step 4: Final Polar Equation

The equation \(r\cos\theta = 5\) is the polar equation equivalent to the Cartesian equation \(x = 5\). This represents the same vertical line in polar coordinates.

Key Concepts

Cartesian Coordinates

Cartesian coordinates form a rectangular coordinate system used to locate points in a plane through two perpendicular axes, usually labeled as 'x' and 'y'. Each point in the plane can be specified by a pair of numerical coordinates, which are the distances to the point from two fixed perpendicular directed lines, measured in the same unit of length. The point where the axes intersect is known as the origin.

• The x-coordinate represents the horizontal position.
• The y-coordinate represents the vertical position.

This system is especially useful for defining locations in familiar two-dimensional space and plotting straightforward vertical and horizontal lines, such as the line represented by the equation \(x = 5\). In this case, \(x = 5\) implies all points are positioned at a constant horizontal distance of 5 units from the y-axis, producing a vertical line.

Polar Coordinates

Polar coordinates offer a different method of representing points in a plane, using two different values: the radial distance from a fixed point known as the origin, and the angle formed with respect to a reference direction, usually the positive x-axis. This system is often more convenient for dealing with symmetrical patterns and angles, or when expressing complex shapes.

• The radial coordinate, denoted as 'r', signifies the distance from the origin.
• The angular coordinate, denoted as '\(\theta\)', is the angle formed with the positive x-axis.

These coordinates are particularly handy in circular and rotational problems, as they allow for easier calculations of angles and distances compared to Cartesian points. Unlike Cartesian coordinates, polar coordinates can contribute unique solutions to geometric representations.

Coordinate Conversion

Converting between Cartesian and Polar coordinates is a vital skill in mathematics, as each system offers unique advantages in different scenarios. The main formulas for conversion are:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(r = \sqrt{x^2 + y^2}\)
• \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)

These equations enable us to translate a point's location from the Cartesian plane to the polar plane, or vice versa, preserving their geometric relationships. For instance, transforming a Cartesian line, like \(x = 5\), into its polar form becomes straightforward when applying these substitutions, leading to the equivalent polar form \(r \cos \theta = 5\). This ensures that the same geometric features are represented in different coordinate systems.

Vertical Line in Polar Form

Re-interpreting a vertical line from Cartesian coordinates into polar form requires an understanding of how lines are expressed differently across systems. A vertical line like \(x = 5\) remains a constant distance from the y-axis in the Cartesian system. When translated into polar form, the same line is articulated through the equation \(r \cos \theta = 5\).

• This equation states that for any angle \(\theta\), the component of the radial distance 'r' in the x-direction remains constant at 5 units.
• The polar coordinate system allows for all possible angles \(\theta\), capturing the essence of a vertical line in its entirety.

This translation showcases the versatility of polar coordinates in representing objects and shapes that have traditionally been viewed through the rectangular lens of Cartesian planes. So, while it may initially seem complex, this conversion maintains the verticality of the line, allowing it to be used in broader mathematical and physics problems.