Question

Find parametric equations for the following curves. Include an interval for the parameter values. The lower half of the circle centered at (-2,2) with radius \(6,\) oriented in the counterclockwise direction

Short answer

Answer: The parametric equations for the lower half of the circle are x(t) = -2 + 6*cos(t) and y(t) = 2 + 6*sin(t), with the parameter values for t in the interval [0, π].

Step-by-step solution

Find the equation of the circle

Given the center of the circle is (-2,2) and the radius is 6, we can find the equation by using the formula (x-a)^2 + (y-b)^2 = r^2, where (a,b) is the center and r is the radius. Thus, the equation of the circle would be (x-(-2))^2 + (y-2)^2 = 6^2, which simplifies to (x+2)^2 + (y-2)^2 = 36.

Decide the range for the parameter

Since we have the lower half of the circle, and it is oriented counterclockwise, we will choose our parameter, say t, to be the angle formed by the horizontal line and the line connecting the center and a point on the circle. When t=0, the circle will be at the leftmost point, and when t=π, it will be at the rightmost point. Therefore, we can take our parameter values within the interval [0, π].

Convert the equation into parametric form

To convert the equation of the circle to parametric form, we can use the following relations: x = a + r*cos(t) y = b + r*sin(t) Here, we will substitute the values of a, b, and r as follows:y x = -2 + 6*cos(t) y = 2 + 6*sin(t) So, the parametric equations for the lower half of the circle centered at (-2,2) with radius 6 with counterclockwise orientation are: x(t) = -2 + 6*cos(t) y(t) = 2 + 6*sin(t) With the interval for the parameter values t as [0, π].

Key concepts

Circle Equations

To understand parametric equations, we first need to grasp the concept of circle equations. A circle in an XY-plane can be described using its center point and radius. The general equation for a circle centered at \(a,b\) with radius \(r\) is:
\[(x-a)^2 + (y-b)^2 = r^2\]
This equation is a representation of all points \(x, y\) that lie on the circle. In the given exercise, the circle's center is at \((-2,2)\) and its radius is 6, leading to the equation:
\[(x+2)^2 + (y-2)^2 = 36\]
This equation encompasses all points on the circle. However, when dealing with portions of a circle, like just the lower half, further adjustments are needed.

Parametrization

Parametrization is a method to represent a set of values, such as a curve, using parameters, usually involving trigonometric functions for circular paths. In the context of circle equations, you can express the position on the circle using the angle \(t\) formed with a reference line. Typically, parametric equations for circles use the relations:

• \(x = a + r\cos(t)\)
• \(y = b + r\sin(t)\)

In the given problem, where only the lower half of the circle is considered, the parameter \(t\) ranges from \(0\) to \(\pi\). When \(t = 0\), the point is at the leftmost side of the circle, and at \(t = \pi\), the point is at the rightmost side.
Thus, our equations become:

• \(x(t) = -2 + 6\cos(t)\)
• \(y(t) = 2 + 6\sin(t)\)

This representation allows us to easily map out the lower semicircle's path by varying \(t\) within the specified interval.

Trigonometric Functions

Trigonometric functions are fundamental in converting circle equations into parametric form. They are based on the relationships between the angles and sides of right-angled triangles. For a circle, these functions relate the angle with horizontal and vertical distances from the center:

• Cosine ( \cos(t)) provides the horizontal distance from the circle's center.
• Sine ( \sin(t)) provides the vertical distance from the circle's center.

In parametric equations, these trigonometric functions help describe every point on the circle's circumference. For example, using \(t\) as an angle, \(\cos(t)\) and \(\sin(t)\) yield x and y coordinates:

• \(x = -2 + 6\cos(t)\)
• \(y = 2 + 6\sin(t)\)

These coordinates, controlled by the parameter \(t\), let you trace the circle's shape. Trigonometric functions thus provide a concise way to analyze and graph geometrical paths like circles.