Problem 32 - Essential Calculus

Question

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

Step-by-Step Solution

Answer

The standard equation of the circle in the rectangular coordinate system is \( (x - h)^2 + (y - k)^2 = r^2 \).

Step 1: Express \( \cos \theta \) and \( \sin \theta \) in terms of \( x \) and \( y \)

Rearrange the given equations to express \( \cos \theta \) and \( \sin \theta \) in terms of \( x \) and \( y \) respectively.\n From \( x = h + r \cos \theta \), we get \( \cos \theta = \frac{x - h}{r} \). \n Similarly, from \( y = k + r \sin \theta \), we get \( \sin \theta = \frac{y - k}{r} \).

Step 2: Use the Pythagorean identity

Substitute \( \cos \theta = \frac{x - h}{r} \) and \( \sin \theta = \frac{y - k}{r} \) into the Pythagorean trigonometric identity, \( \sin^2 \theta + \cos^2 \theta = 1 \).\n This gives: \( \left(\frac{x - h}{r}\right)^2 + \left(\frac{y - k}{r}\right)^2 = 1 \).

Step 3: Convert to standard form

The expression obtained in Step 2 is very close to the standard form. To get the exact standard form of the equation of a circle, we simply have to clear the denominator \( r^2 \) in the left hand side:\n This results in: \( (x - h)^2 + (y - k)^2 = r^2 \).\n This is the standard form of the equation of a circle in the rectangular coordinate system, where \( (h, k) \) are the coordinates of the center of the circle and \( r \) is the radius.

Key Concepts

Circle Equation

Understanding the circle equation is crucial when dealing with problems involving circles in mathematics. The standard form of a circle's equation is \[ (x - h)^2 + (y - k)^2 = r^2 \] Here, \((h, k)\) represents the center of the circle, and \(r\) is the circle's radius. The equation defines all of the points \((x, y)\) that lie on the perimeter of the circle.
If you start with parametric equations like \(x = h + r \cos \theta\) and \(y = k + r \sin \theta\), it is possible to eliminate the parameter \(\theta\) and derive the circle's equation in its standard form.
The key steps involve expressing \(\cos \theta\) and \(\sin \theta\) in terms of \(x\) and \(y\) and then using trigonometric identities to remove the parameter.

Trigonometric Identities

Trigonometric identities are powerful tools in mathematics, especially when working with parametric equations. In the case of converting parametric equations of a circle to its standard form, the Pythagorean identity plays a crucial role. This identity is \[ \sin^2 \theta + \cos^2 \theta = 1 \]This equation holds for any angle \(\theta\).
To link parametric equations with the standard circle equation, you express \(\cos \theta\) as \(\frac{x - h}{r}\) and \(\sin \theta\) as \(\frac{y - k}{r}\).
When you substitute these into the Pythagorean identity, you derive:

• \(\left(\frac{x - h}{r}\right)^2 + \left(\frac{y - k}{r}\right)^2 = 1 \)

Clearing the fractions ultimately leads to the circle equation in standard form. This process shows how trigonometric identities help bridge different representations of the same geometric shape.

Rectangular Coordinates

Rectangular coordinates, commonly known as Cartesian coordinates, are a classic way to represent points in a plane. They use a grid system with a horizontal \(x\)-axis and a vertical \(y\)-axis. Each point in the plane is identified by its \(x\) and \(y\) values which denote its horizontal and vertical distances from the origin \((0,0)\), respectively.
When dealing with circles, rectangular coordinates simplify how we understand and solve for the points that form a circle.
The transformation from parametric equations to the circle's equation in rectangular coordinates involves both algebraic manipulation and trigonometric insights.

• The parametric equations represent points on a circle using parameters, like angle \(\theta\).
• The goal is to eliminate this parameter and express the relationship directly in terms of \(x\) and \(y\).
• This gives us a clear geometric interpretation: the resulting equation describes a circle centered at \((h, k)\) with radius \(r\).

Rectangular coordinates provide a straightforward visualization, especially when plotting circles or any geometric figures on a graph.