Question

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The lower half of the circle centered at $(-2,2)$ with radius 6 oriented in the counterclockwise direction

Short answer

Question: Find the parametric equations for the lower half of a circle centered at (-2, 2) with a radius of 6 oriented in the counterclockwise direction. Include the parameter interval. Answer: The parametric equations for the lower half of the circle are: x(t) = -2 + 6cos(t), y(t) = 2 + 6sin(t), where t is in the interval [π, 2π].

Step-by-step solution

Write the general form of parametric equations for a circle

A general form of parametric equations for a circle with radius $r$ centered at $(h,k)$ can be expressed as follows: $x(t)=h+r\cos (t),$ $y(t)=k+r\sin (t),$ In our case, the center of the circle is $(-2,2)$, and the radius is 6.

Substitute the center coordinates and radius into the parametric equations

In our exercise, the center of the circle is at $(-2,2)$, and the radius, $r$, is 6. We can substitute these values into the parametric equations to get: $x(t)=-2+6\cos (t),$ $y(t)=2+6\sin (t).$

Find the interval for the parameter

To create parametric equations for the lower half of the circle, we need the parameter $t$ to start at the point where the counterclockwise direction would initiate the lower half (i.e., the left-most point of the circle) and end at the point where the lower half of the circle completes (i.e., the right-most point of the circle). Since the circle's center is at the point $(-2,2)$, the left-most point of the circle is at $(-2-6,2)=(-8,2)$. Similarly, the right-most point is $(0,2)$. At point $(-8,2)$, the angle $t$ equals $\pi$ or ${180}^{\circ }$, and at point $(0,2)$, the angle $t$ equals $2\pi$ or ${360}^{\circ }$. However, we want to find parametric equations for the lower half of the circle only, which means we need values of $t$ between $\pi$ and $2\pi$. Thus, the interval for the parameter $t$ is $[\pi ,2\pi ]$.

Combine the parametric equations and parameter interval

Now, we have the parametric equations and the interval for the parameter $t$. We can write them together as our final answer: Parametric Equations for the lower half of the circle: $x(t)=-2+6\cos (t),$ $y(t)=2+6\sin (t),$ where $t\in [\pi ,2\pi ]$.

Key concepts

Parametric Equations

In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters. For curves, these equations express the coordinates of the points on the curve as functions of a single parameter. Parametric representations are particularly useful for defining curves that do not represent functions in the traditional sense, as they allow a multi-valued relationship. For example, the equation of a circle cannot be expressed as a single function in the standard form y = f(x), because for some x values there are two corresponding y values. Thus, parametric equations are highly suitable for describing circular motion and other complex geometries.

In the context of circles, the general parametric equations are given as:

• $x(t)=h+r\cos (t)$,
• $y(t)=k+r\sin (t)$,

where $h$ and $k$ are the coordinates of the circle’s center, $r$ is the radius, and $t$ is the parameter, often related to time or angle in trigonometric applications. Students should pay close attention to how the center point and the radius influence the form of these equations, as manipulation of these values will directly affect the shape and position of the resulting curve.

Circle Equations

Circle equations typically come in two forms: the standard form and the general form. The standard form of a circle's equation with center $(h,k)$ and radius $r$ is $(x-h{)}^2+(y-k{)}^2=r^2$.This form highlights the symmetry of a circle and its radius-centered properties. However, when dealing with motion along a circle or when needing a more dynamic representation, parametric equations are preferred. The parameters in the previous section facilitate the description of any point along the circumference via trigonometric functions. Understanding how to navigate between the standard and parametric forms of the circle equation provides students with a powerful tool for tackling a wide range of geometric and motion problems.

Trigonometric Functions

Trigonometric functions are fundamental in understanding and expressing the relationships between angles and side lengths in right triangles. Their usefulness extends far beyond triangles, as they are crucial in modeling periodic phenomena, such as waves and circular motion. In our example with the parametric equations for a circle, the cosine and sine functions describe how a point moves around the circumference.

The functions $\cos (t)$ and $\sin (t)$ respectively represent the x and y coordinates of a unit circle as a function of the angle $t$. It's essential for students to be comfortable with these functions, recognizing that they describe a circle's shape due to their periodic nature. They will also need to understand how these functions correlate to the specific quadrants of a circle and the sign changes associated with them when the angle parameter varies.

Interval for Parameter Values

The interval for parameter values determines the portion of the curve described by parametric equations that is of interest. In the context of circle parametric equations, $t$ is frequently used to represent the angle made with the positive x-axis, making it a natural parameter for circular motion. By restricting the interval of $t$, one can selectively describe only a certain arc or sector of a circle.

In the given example of the lower half of a circle, the interval $[\pi ,2\pi ]$ ensures that only the points from the left-most direct position in a counterclockwise direction down to the right-most point are considered. This interval encapsulates the essence of the lower half-circle's journey. Students should appreciate the significance of the chosen interval, as it directly impacts the representation of the curve in question, enabling precise mathematical modeling of real-life situations.