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Chapter 11: Problem 43

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.

Short Answer

Expert verified
The parametrization is \(x = 2 - \cos(\theta)\), \(y = \sin(\theta)\) for \(\theta\) starting at \(\pi\).

Step by step solution

01

Understand the Circle's Equation

The given equation is \((x-2)^2 + y^2 = 1\). This represents a circle centered at \((2, 0)\) with a radius of \(1\). Our task is to describe a method to traverse the circumference of this circle.
02

Determine Standard Parameterization

For a circle centered at \((h, k)\) with radius \(r\), the standard parameterization using angle \(\theta\) is \((h + r\cos(\theta), k + r\sin(\theta))\) where \(\theta\) is in radians. Given \(h = 2\), \(k = 0\), and \(r = 1\), the standard parameterization is \((2 + \cos(\theta), \sin(\theta))\).
03

Adjust for Clockwise Direction

To move clockwise around the circle, we must reverse the direction of \(\theta\)'s rotation. Hence, replace \(\theta\) with \(-\theta\). The new parameterization is \(x = 2 + \cos(-\theta) = 2 + \cos(\theta)\) and \(y = \sin(-\theta) = -\sin(\theta)\).
04

Verify the Starting Point

Substitute \(\theta = 0\) into the parameterization. This gives \(x = 2 + \cos(0) = 3\) and \(y = -\sin(0) = 0\). However, we need it to start at \((1,0)\). Hence, adjust \(\theta\) so that the starting point aligns, using \(\theta = \pi\). Check, at \(\theta = \pi\), \(x = 2 + \cos(\pi) = 1\) and \(y = -\sin(\pi) = 0\), which correctly gives the starting point.
05

Final Parametrization

By rotating starting from \(\theta = \pi\), the parametrization becomes \(x = 2 + \cos(\theta + \pi)\) and \(y = -\sin(\theta + \pi)\), ensuring it starts at \((1,0)\) and moves clockwise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circle Equation
In mathematics, the circle equation provides a concise way to describe all points that are a fixed distance, known as the radius, from a central point, called the center. The general form of the circle equation is
  • egin{equation} (x - h)^2 + (y - k)^2 = r^2 ag{Circle Equation} dots
where
  • \( (h, k) \) is the center of the circle
  • \( r \) is the radius.
For our specific problem, we have
  • \( (x-2)^2 + y^2 = 1 \).
This indicates a circle centered at
  • \( (2,0) \).
The radius is
  • \( 1 \), meaning all points on this circle are exactly one unit away from the center.
Clockwise Rotation
When discussing movement along a circle, the direction of rotation is crucial. Typically, the angle
  • \( \theta \)
is measured counterclockwise from the positive x-axis. However, in this exercise, we want the motion to be clockwise. To achieve this, we reverse the direction of
  • \( \theta \)
by replacing it with
  • \( -\theta \).
**Clockwise Direction Characteristics**:
  • It starts at a given point, here,\( (1,0) \).
  • The parametric angle changes from positive (counterclockwise) to negative (clockwise).
The clockwise rotation specifically demands adjusting the starting angle to ensure the path starts and rotates correctly around the circle.
Parameterization Technique
Parametrization involves creating a set of equations that captures all the points on a curve using a parameter, typically an angle in circles. A standard approach to parametrize a circle is to use trigonometric functions, sine and cosine, because they naturally describe circular motion. In standard form, for a circle centered at
  • \( (h, k) \)
with radius
  • \( r \)
, the parametrization is:
  • \( x = h + r \cos(\theta) \)
  • \( y = k + r \sin(\theta) \)
In our case, with specific values of
  • \( h = 2, k = 0 \),
  • and \( r = 1 \),
the parameterization becomes
  • \( x = 2 + \cos(\theta) \)
  • \( y = \sin(\theta) \).
However, adjustments are needed to ensure beginning at
  • \( (1,0) \)
and rotating clockwise.
Trigonometric Functions
Trigonometric functions are fundamental when parameterizing circles, especially sine and cosine functions, which relate angles to circle points. **Key Functions Used: **
  • **Cosine** (\( \cos \)) determines the x-coordinate in the parametric equation.
  • **Sine** (\( \sin \)) determines the y-coordinate.
**Properties of Sine and Cosine:**
  • They both have a range from -1 to 1, matching the demands of a circle's unit distance.
  • They are periodic with a period of \( 2\pi \), meaning the pattern repeats every full rotation.
To account for a clockwise rotation, the angle
  • \( \theta \)
is expressed as
  • \( -\theta \),
reversing the normal direction and allowing the circle to be traced in the opposite direction. This adjustment ensures the trigonometric functions align correctly with the circle's desired movement.

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Most popular questions from this chapter

Give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes. \(y^{2}-x^{2}=1,\) left 1, down 1

Find the center, foci, vertices, asymptotes, and radius, as appropriate, of the conic sections. \(2 x^{2}-y^{2}+6 y=3\)

The curve with parametric equations $$ x=(1+2 \sin \theta) \cos \theta, \quad y=(1+2 \sin \theta) \sin \theta $$ is called a lima莽on and is shown in the accompanying figure. Find the points \((x, y)\) and the slopes of the tangent lines at these points for $$a. \theta=0 . \quad b. \theta=\pi / 2 . \quad c. \theta=4 \pi / 3$$

The length of the curve \(r=f(\theta), \alpha \leq \theta \leq \beta\) Assuming that the necessary derivatives are continuous, show how the substitutions $$x=f(\theta) \cos \theta, \quad y=f(\theta) \sin \theta$$ (Equations 2 in the text) transform $$L=\int_{\alpha}^{\beta} \sqrt{\left(\frac{d x}{d \theta}\right)^{2}+\left(\frac{d y}{d \theta}\right)^{2}} d \theta$$ into $$L=\int_{\alpha}^{\beta} \sqrt{r^{2}+\left(\frac{d r}{d \theta}\right)^{2}} d \theta$$

Give equations for parabolas and tell how many units up or down and to the right or left each parabola is to be shifted. Find an equation for the new parabola, and find the new vertex, focus, and directrix. \(y^{2}=-12 x, \quad\) right 4, up 3
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