Problem 29 - Calculus Early Transcendentals

Question

Find parametric equations for the following circles and give an interval for the parameter values. Graph the circle and find a description in terms of \(x\) and \(y .\) Answers are not unique. A circle centered at \((2,3)\) with radius \(1,\) generated counterclockwise

Step-by-Step Solution

Answer

Question: Find the parametric equations of a circle centered at (2, 3) with a radius of 1, generated counterclockwise. Determine an interval for the parameter values, graph the circle, and provide a description of the circle in terms of x and y. Answer: The parametric equations are \(x(t) = 2 + \cos t\) and \(y(t) = 3 + \sin t\). The interval for the parameter values is \(0 \le t \le 2\pi\). The graph shows a circle centered at (2, 3) with a radius of 1, generated counterclockwise. The description of the circle in terms of x and y is \((x - 2)^2 + (y - 3)^2 = 1\).

Step 1: Finding the Parametric Equations

To find the parametric equations for the circle, we can use the following formula: \(x(t) = a + r \cos t\) \(y(t) = b + r \sin t\) where \((a, b)\) is the center of the circle and \(r\) is the radius. In this case, the circle is centered at \((2, 3)\) and has a radius of 1. So our parametric equations will be: \(x(t) = 2 + 1 \cos t = 2 + \cos t\) \(y(t) = 3 + 1 \sin t = 3 + \sin t\)

Step 2: Identifying the Interval for Parameter Values

Since the problem uses a counterclockwise generation, we will use a positive angle for the parameter \(t\). We can start at \(t = 0\) and go around the circle once to have a complete representation of the circle. This will give us the interval: \(0 \le t \le 2\pi\)

Step 3: Graphing the Circle

Using our parametric equations, \(x(t) = 2 + \cos t\) and \(y(t) = 3 + \sin t\), we can graph the circle. The graph will show a circle centered at \((2, 3)\) with a radius of 1, generated counterclockwise.

Step 4: Finding the Description in terms of \(x\) and \(y\)

To find a description of the circle in terms of \(x\) and \(y\), we can use our parametric equations and convert them to Cartesian coordinates: \((x - 2)^2 + (y - 3)^2 = r^2\) Since the radius is 1, we can write the equation as: \((x - 2)^2 + (y - 3)^2 = 1\) This is the required description of the circle in terms of \(x\) and \(y\).

Key Concepts

Parametric Equations

Imagine drawing a shape with a pen without lifting it off the paper; parametric equations are the mathematical representation of this motion. They express the coordinates of the points that make up a curve using a parameter, often denoted as 't'. Consider parametric equations like a set of instructions: for each value of 't', they indicate a specific location in the coordinate system.

For circles, the parametric equations involve sine and cosine functions. The equations \(x(t) = a + r \cos t\) and \(y(t) = b + r \sin t\) describe a circle with a radius 'r' centered at point \(a, b\). The parameter 't' typically represents an angle in radians. As 't' varies, the point \(x(t), y(t)\) traces out the circle. This brings us to our example, where the circle has a center at \(2,3\) and a radius of 1. The parametric equations are \(x(t) = 2 + \cos t\) and \(y(t) = 3 + \sin t\).

The key here is to envision how the cosine and sine functions oscillate. As 't' increases, the point on the circle moves according to these oscillations, forming a smooth, circular path.

Circle Graphing

Graphing a circle from its parametric equations can feel a bit like connecting the dots, but with a continuous motion. Each pair of \(x(t)\) and \(y(t)\) values plots a single point on the plane, and together, they form the shape of the circle.

To graph our example circle, we take different values of 't' from the interval \(0 \le t \le 2\pi\), calculate the corresponding \(x\) and \(y\) using our parametric equations, and plot these points. Since the problem specifies a counterclockwise direction, we can imagine 't' starting from the rightmost point of the circle and sweeping around to meet the starting point again. The result is a neatly traced circle centered at \(2,3\) with a radius of 1. It's important to get comfortable with this process because graphing helps provide a visual understanding of parametrically defined curves.

Conversion to Cartesian Coordinates

Converting parametric equations to Cartesian coordinates is a vital skill. It unifies the parametrically described shapes into the more traditional Cartesian plane. For circles, this conversion leads to the equation \((x - a)^2 + (y - b)^2 = r^2\), where \(a\) and \(b\) are the coordinates of the circle's center and 'r' is its radius.

In the step-by-step solution of our circle problem, we actually performed this transformation. The circle that was initially described by the parametric equations \(x(t) = 2 + \cos t\) and \(y(t) = 3 + \sin t\) was later expressed in Cartesian form as \( (x - 2)^2 + (y - 3)^2 = 1\), after recognizing that the radius is 1. The ability to switch between parametric and Cartesian equations allows for a deeper understanding of the underlying geometry and enhances flexibility in problem-solving.

Trigonometric Functions

The use of trigonometric functions in parametric equations, especially for circles, is rooted in their fundamental properties. Sine \(\sin t\) and cosine \(\cos t\) represent the vertical and horizontal coordinates, respectively, of a point on the unit circle as it moves counter-clockwise starting from the right-most point.

In our example, the parametric equations include \(\cos t\) for the \(x\)-coordinate and \(\sin t\) for the \(y\)-coordinate adjustments. With these functions, as 't' increases from 0 to \(2\pi\), the point \(x(t), y(t)\) travels once around the circle. These functions are periodic, so they repeat their values in a consistent pattern, further reinforcing the circular path when graphed against the parameter 't'. Understanding how these trigonometric functions work is crucial not just for circles but for many areas of mathematics where periodicity and angles are involved.