Problem 33 - Thomas Calculus

Question

Find a parametrization for the circle \((x-2)^{2}+y^{2}=1\) starting at \((1,0)\) and moving clockwise once around the circle, using the central angle \(\theta\) in the accompanying figure as the parameter.

Step-by-Step Solution

Answer

Parametrize with \(x = 2 + \cos(\pi + \theta), y = -\sin(\pi + \theta), \theta \in [0, 2\pi]\).

Step 1: Identify Circle Equation Parameters

The given circle equation is \[(x-2)^2 + y^2 = 1.\]This implies that the circle has its center at \((2, 0)\) and radius \(1\).

Step 2: Determine Standard Parametric Equations

A standard parametrization for a circle with center \((h, k)\) and radius \(r\) is \[x = h + r \cos(\theta) \y = k + r \sin(\theta).\]In our case, \(h=2\), \(k=0\), and \(r=1\). Substituting these into the standard formula, we get:\[x = 2 + \cos(\theta) \y = \sin(\theta).\]

Step 3: Adjust for Clockwise Motion

Since the problem specifies a clockwise parameterization, we need to reverse the standard trigonometric functions:\[x = 2 + \cos(\theta) \y = - \sin(\theta).\]This change makes the circle traverse clockwise as \(\theta\) increases.

Step 4: Check Parameterization Start Point

For \(\theta = 0\), the point is:\[x = 2 + \cos(0) = 3 \y = - \sin(0) = 0.\]We must find where \((1, 0)\) lies. Adjusting \(\theta\) so that \(x = 1\) results in\[\cos(\theta) = 1 - 2 = -1.\]This happens at \(\theta = \pi\), so the parametrization is \[x = 2 + \cos(\pi + \theta), \y = - \sin(\pi + \theta).\]

Step 5: Verify the Solution and Parameter Range

Calculating for \(\theta = \pi\) in parameterization,\[x = 2 + \cos(2\pi) = 1 \y = - \sin(2\pi) = 0.\]Hence, it starts at \((1, 0)\). The parameter \(\theta\) should range from \(0\) to \(2\pi\) to complete one full clockwise rotation around the circle.

Key Concepts

Clockwise Motion

In the realm of mathematics, understanding the direction of motion around a circle is important. Typically, parametric equations of a circle use trigonometric functions that describe a counterclockwise motion.

However, when the task involves moving clockwise, we must adjust these functions. In standard form, the parameterization for counterclockwise motion involves

• \(x = h + r \cos(\theta)\)
• \(y = k + r \sin(\theta)\)

To achieve clockwise movement, you swap the sign of the sine function. Why? Because while cosine remains unchanged, the sine function determines vertical positioning.

By reversing the sign, you effectively move in the opposite direction around the circle. Thus, the clockwise parameterization becomes:

• \(x = h + r \cos(\theta)\)
• \(y = k - r \sin(\theta)\)

This adjustment ensures that as \(\theta\) increases, the motion follows a clockwise path.

Circle Equation

A circle equation in its standard form is \((x-h)^2 + (y-k)^2 = r^2\), where:

• \((h, k)\) is the center of the circle
• \(r\) represents the radius of the circle

For instance, in \((x-2)^2 + y^2 = 1\), the circle is centered at \((2, 0)\) with a radius of 1.This form of circle equation is essential as it directly describes the position and size of the circle in a coordinate plane.

Knowing this allows us to find parametric equations that represent the same circle, enabling us to map specific points or directions with precise calculations.

Trigonometric Functions

Trigonometric functions are key to translating angles into coordinates along a circle.

In parametrizing a circle, we predominantly use cosine and sine functions:

• \(\cos(\theta)\) represents horizontal displacement
• \(\sin(\theta)\) represents vertical displacement

The standard usage of these functions lets us express any point on a circle of radius \(r\) centered at \((h, k)\) through:

• \(x = h + r \cos(\theta)\)
• \(y = k + r \sin(\theta)\)

Understanding trigonometric functions in these terms allows for a seamless conversion of angles into positional data, facilitating tasks such as plotting motion or specifying points along the circle's path.

Adjusting the sign of sine to negative, as done in clockwise motion, leverages these functions effectively to reverse travel direction.