Problem 49 - Advanced Engineering Mathematics

Question

Convert the given equation to rectangular cocrdinates. $$ r=5 \sec \theta $$

Step-by-Step Solution

Answer

The rectangular coordinate equation is \( x = 5 \).

Step 1: Understand Polar Coordinates Terminology

The equation is in polar coordinates, which typically involve the variables \( r \) and \( \theta \). Here, \( r \) is the distance from the origin and \( \theta \) is the angle from the positive x-axis. The given equation is \( r = 5 \sec \theta \).

Step 2: Express \( \sec \theta \) in terms of Trigonometric Identity

Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Substituting this in the equation gives us \( r = \frac{5}{\cos \theta} \).

Step 3: Rewrite \( r \) in Terms of \( x \, \text{and} \, y \)

In polar to rectangular conversion, \( x = r \cos \theta \) and \( y = r \sin \theta \). Since \( \cos \theta = \frac{x}{r} \), we substitute it in our equation: \( r = \frac{5}{\frac{x}{r}} = \frac{5r}{x} \).

Step 4: Solve for \( x \) in Terms of \( r \)

After simplifying \( r = \frac{5r}{x} \), we can multiply both sides by \( x \) to get \( x = 5 \).

Step 5: Conversion Verification

Check if \( x = 5 \) is consistent with the polar form by substituting into the equations \( x = r \cos \theta \): \( 5 = r \cos \theta \). Given \( r = 5 \sec \theta \) implies \( r \cos \theta = 5 \), confirms that \( x = 5 \) is correct.

Key Concepts

Polar Coordinates

Polar coordinates provide a unique system for locating points on a plane using the radius and angle. This system is particularly useful when dealing with circular or rotational systems.
In polar coordinates:

• \( r \) represents the radial distance from the origin (or the center of the circle).
• \( \theta \) (theta) is the angle made with the positive x-axis.

Polar coordinates are often used in scenarios where Cartesian (rectangular) coordinates are less convenient, such as when problems have symmetry about a point or involve rotation. In our exercise, the polar equation \( r = 5 \sec \theta \) shows how this circular system relates distance \( r \) to the angle \( \theta \). This understanding is key to grasping how the conversion to rectangular coordinates reveals a straight line: \( x = 5 \).

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, represent points on a plane using two perpendicular axes—horizontal (x-axis) and vertical (y-axis).
In rectangular coordinates:

• \( x \) describes the horizontal distance from the origin.
• \( y \) describes the vertical distance from the origin.

The conversion from polar to rectangular coordinates involves relating \( r \) and \( \theta \) to \( x \) and \( y \):

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

These formulas stem from basic trigonometric definitions: the cosine of an angle in a right triangle relates the adjacent side (x) to the hypotenuse (r), and the sine relates the opposite side (y) to the hypotenuse (r).
This exercise emphasizes the simplicity of such conversions once you know \( r \) and \( \theta \) and need an equivalent \( x, y \) description. The polar equation \( r = 5 \sec \theta \) converts neatly to \( x = 5 \), a line parallel to the y-axis.

Trigonometric Identities

Trigonometric identities are mathematical relationships expressing the connections between various trigonometric functions like sine, cosine, and tangent and their reciprocals—cosecant, secant, and cotangent.
A key identity in this exercise is the secant function:

• \( \sec \theta = \frac{1}{\cos \theta} \)

This identity transforms our given polar equation \( r = 5 \sec \theta \) into a more tractable form: \( r = \frac{5}{\cos \theta} \).
Understanding these connections allows us to replace \( \sec \theta \) with expressions involving \( \cos \theta \), facilitating the shift from polar to rectangular equations. The expression \( r = \frac{5}{\cos \theta} \) aids in determining the rectangular coordinate by letting \( x = 5 \), as shown by solving \( r \cos \theta = 5 \).
Trigonometric identities not only support conversions but also streamline solving problems across different coordinate systems, making them indispensable tools in mathematics.