Problem 71 - Advanced Engineering Mathematics

Question

Convert the given equation to rectangular coordinates. $$ \rho=10 $$

Step-by-Step Solution

Answer

The rectangular equation is \( x^2 + y^2 = 100 \).

Step 1: Understand the Polar Equation

The equation given is in polar coordinates, \[ \rho = 10 \]where \( \rho \) represents the distance from the origin to the point.

Step 2: Recall Conversion Formulas

In polar coordinates, any point can be converted to rectangular coordinates using the formulas:\[ x = \rho \cos(\theta) \]\[ y = \rho \sin(\theta) \]

Step 3: Substitute \( \rho = 10 \) in Rectangular Conversion

Substitute \( \rho = 10 \) into the conversion formulas for \( x \) and \( y \):\[ x = 10 \cos(\theta) \]\[ y = 10 \sin(\theta) \]

Step 4: Square and Add the Conversion Equations

Square both equations and add them:\[ x^2 = (10 \cos(\theta))^2 \]\[ y^2 = (10 \sin(\theta))^2 \]\[ x^2 + y^2 = 100(\cos^2(\theta) + \sin^2(\theta)) \]

Step 5: Apply Trigonometric Identity

Apply the Pythagorean identity:\[ \cos^2(\theta) + \sin^2(\theta) = 1 \]Thus,\[ x^2 + y^2 = 100 \]

Step 6: Write the Rectangular Equation

The equation in rectangular form is:\[ x^2 + y^2 = 100 \]This represents a circle with radius 10 centered at the origin.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative coordinate system to the more commonly used rectangular coordinates. They are particularly useful in scenarios where points are naturally described by their distance and angle from a fixed point, known as the pole (which corresponds to the origin in rectangular coordinates). In polar coordinates, a point is defined by the radial distance \( \rho \) from the pole and the angle \( \theta \) from the positive x-axis.

This system is visually represented by concentric circles (representing various \( \rho \) values) around the origin and rays extending out from the origin at different angles \( \theta \). When given an equation in polar coordinates like \( \rho = 10 \), it implies that each point on the circle is exactly 10 units away from the pole, making it simple to interpret geometrically. Polar coordinates are widely used in fields involving circular or rotational motion like physics and engineering arenas.

Coordinate Conversion

Coordinate conversion is the process of translating coordinates from one system to another, which, in this context, refers to converting polar coordinates to rectangular coordinates. This conversion is crucial when you wish to integrate equations or points between systems that have inherent advantages based on the problem's nature.

The primary formulas used for conversion from polar to rectangular coordinates are:

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

These formulas derive from basic trigonometry and the definition of a right triangle. Here, \( \rho \) acts as the hypotenuse of the triangle, while the \( x \) and \( y \) components are the adjacent and opposite sides respective to the angle \( \theta \). By substituting given values into these formulas, such as \( \rho = 10 \), one can easily transition between describing a point in polar form to rectangular form.

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that simplify expressions and solve equations involving trigonometric functions. One essential identity used in coordinate conversion is the Pythagorean identity: \( \cos^2(\theta) + \sin^2(\theta) = 1 \). This identity reflects a key relationship between the sine and cosine functions that holds for any angle \( \theta \).

In our exercise, applying the Pythagorean identity allows us to transform expressions involving trigonometric components into a familiar geometric form. By using the identity, the sum \( 100(\cos^2(\theta) + \sin^2(\theta)) \) simplifies to 100, confirming the equation \( x^2 + y^2 = 100 \). This is the equation of a circle in rectangular coordinates, with a radius derived from the original \( \rho \) value in the polar system.

Understanding these identities assists in bridging the gap between the polar and rectangular systems, making complex problems more approachable and solvable.