Problem 53 - Thomas Calculus

Question

Replace the Cartesian equations in Exercises \(53-66\) with equivalent polar equations. $$x=7$$

Step-by-Step Solution

Answer

The equivalent polar equation is \(r \cos\theta = 7\).

Step 1: Understand the Cartesian Equation

The Cartesian equation given is \(x=7\). This represents a vertical line on the Cartesian coordinate plane, where all points on this line have \(x\) coordinates equal to 7, and \(y\) can be any value.

Step 2: Recall the Conversion Formulas

To convert from Cartesian coordinates \((x, y)\) to polar coordinates \((r, \theta)\), we use the formulas: \(x = r \cos\theta\) and \(y = r \sin\theta\).

Step 3: Set Up Conversion for Given Equation

Substitute the value of \(x\) in terms of polar coordinates into the equation. Our given equation is \(x=7\), so substitute \(r \cos\theta = 7\). This matches the general polar form using Cartesian to polar conversions.

Step 4: Simplify Polar Equation

The equivalent polar equation from our substitution is \(r \cos\theta = 7\). This equation describes the same line in polar coordinates.

Key Concepts

Cartesian Coordinates

In the Cartesian coordinate system, every point is defined by a pair of numerical coordinates. These coordinates are expressed as \(x,y\), corresponding respectively to distances along the x-axis (horizontal) and y-axis (vertical).
To illustrate, consider the line given by the equation \(x = 7\). This equation tells us that all points lying on this line have an x-coordinate of 7, while the y-coordinate can be any real number.

• This forms a vertical line parallel to the y-axis, passing through the point (7, 0) when it crosses the x-axis.
• In Cartesian, lines are often easy to visualize as they are straight paths with uniform direction.

While many find the Cartesian system intuitive and simple for plotting points and lines on a plane, there are scenarios where a different system, like polar coordinates, might offer a clearer perspective.

Coordinate Conversion

The process of converting Cartesian coordinates to polar coordinates involves a set of straightforward formulas. This conversion might be necessary to solve problems or describe figures that are more naturally understood in polar terms.
To make this conversion, you start with the formulas:

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

For the Cartesian equation \(x = 7\), the conversion involves using only the first formula, \(x = r \cos\theta\), to set up the corresponding polar equation.

• Simply replace \(x\) with \(r \cos\theta\) to yield \(r \cos\theta = 7\).
• Solving for \(r\) here isn't necessary, as the equation clearly shows the relation between \(r\) and \(\theta\).

Conversions like this illustrate the geometric descriptions that can be unlocked by transitioning from one coordinate system to another.

Polar Equations

Polar equations are integral for representing figures like circles and spirals. They express the location of points using a radial coordinate \(r\) and an angular coordinate \(\theta\), offering a unique way of describing paths and curves.
In our exercise, the Cartesian line \(x = 7\) is transformed into its polar equivalent \(r \cos\theta = 7\). This polar equation describes the exact same line in terms of polar variables.

• The term \(r\) indicates the distance of any point on the line from the origin, while \(\theta\) gives the angle from the positive x-axis.
• This equation effectively defines a collection of points where the component of \(r\) along the x-axis always equals 7.

Polar equations are thus an alternative but powerful way to view and understand lines and curves, especially in contexts where radial symmetry is present or when working on trigonometric aspects of shapes.