Question

Find parametric equations that describe the given situation. A circle of radius 3 , centered at \((1,1),\) that is traced once counter- clockwise on [0,1] .

Short answer

Parametric equations: \(x(t) = 1 + 3\cos(2\pi t)\), \(y(t) = 1 + 3\sin(2\pi t)\).

Step-by-step solution

Understand the Standard Parametric Equation for a Circle

The standard parametric equation for a circle with radius \( r \) and center \( (h, k) \) is given by:\[ x = h + r \cos(\theta) \ y = k + r \sin(\theta)\]where \( \theta \) is a parameter that varies to trace out the circle.

Determine the Values for Center and Radius

For the given circle, the center is \((1,1)\) and the radius is 3. So, substituting these into the parametric equations yields:\[ x = 1 + 3 \cos(\theta) \ y = 1 + 3 \sin(\theta)\]

Adjust the Parameter for Interval [0,1]

To trace the circle once counter-clockwise as \( t \) varies from 0 to 1, we need to adjust \( \theta \) to trace through one full revolution \( [0, 2\pi] \). Set \( \theta = 2\pi t \). Thus, our equations become:\[ x = 1 + 3 \cos(2\pi t) \ y = 1 + 3 \sin(2\pi t)\]

Write Final Parametric Equations

Therefore, the parametric equations describing the circle traced out counter-clockwise from \( t = 0 \) to \( t = 1 \) are:\[ x(t) = 1 + 3 \cos(2\pi t) \ y(t) = 1 + 3 \sin(2\pi t)\] These equations will accurately describe the motion over the interval \( t \in [0,1] \).

Key concepts

Circle Equation

The equation of a circle is one of the foundational components of geometry. It provides a precise mathematical description of a circle's shape and position. In its standard form, the circle equation involves both the center and the radius. For a circle with center \((h, k)\) and radius \(r\), the basic equation is:

• \((x-h)^2 + (y-k)^2 = r^2\)

This equation alone, however, doesn't tell us how to trace or generate the circle. That's where parametric equations come in handy. They offer a dynamic approach, setting out separate equations for \(x\) and \(y\) using an auxiliary variable often named \(\theta\) or a similar parameter. This transforms the static circle equation into a framework for generating the circle dynamically as the parameter varies, which is particularly useful for animation and simulations.

Trigonometric Functions

Trigonometric functions, like sine and cosine, are deeply intertwined with the circle through the unit circle concept. These functions leverage the unit circle to form the backbone of many mathematical contexts including periodic functions and oscillations. The sine and cosine functions are especially powerful when it comes to creating parametric equations for circles due to their periodic nature.In parametric forms, cosine and sine functions model the path traced out on the Cartesian plane, converting angles directly into coordinates. When modeling a circle, we use the following relationships:

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

Here, the \(\cos(\theta)\) and \(\sin(\theta)\) act much like the \(x\) and \(y\) components of a point on a circle, modulated by the radius \(r\). As \(\theta\) varies, these components trace out the full perimeter of the circle, returning to the origin of the period to create a full loop.

Parameterization

Parameterization gives us the ability to represent more complex geometric figures using a set of simpler, continuous functions. In the context of our circle, parameterization uses a parameter, typically \(\theta\) or \(t\), to express the \(x\) and \(y\) coordinates as functions of this parameter.For a given circle with center \((1,1)\) and radius 3, parameterizing involves setting:

• \(x(t) = 1 + 3 \cos(2\pi t)\)
• \(y(t) = 1 + 3 \sin(2\pi t)\)

Here, the use of \(2\pi t\) ensures that as \(t\) goes from 0 to 1, \(\theta\) goes from 0 to \(2\pi\), covering the full circle counter-clockwise in exactly one cycle. This parameterization effectively ties the geometric properties of the circle to a time-like variable, allowing us to delineate exact positions along the circle's path.