Problem 28 - 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 the origin with radius \(12,\) generated clockwise with initial point \((0,12)\)

Step-by-Step Solution

Answer

Question: Find the parametric equations for a circle centered at the origin with radius 12 and generated clockwise with an initial point at (0,12). Provide the interval for the parameter values, and give a description of the circle in terms of x and y. Answer: The parametric equations for the circle are x = 12cos(-t) and y = 12sin(-t). The interval for the parameter values is -π/2 ≤ t ≤ 3π/2. The description of the circle in terms of x and y is x^2 + y^2 = 144.

Step 1: Parametric Equations for the Circle

For a circle centered at the origin with radius \(R\), the parametric equations are given by \(x=R(\cos(t))\) and \(y=R(\sin(t))\), where \(t\) is the parameter. In our case, the circle has a radius \(R = 12\). Since the circle generates clockwise with an initial point at \((0,12)\), we need to adjust the trigonometric functions accordingly. To get a clockwise rotation, we can invert the cosine function (since it's usually anticlockwise). So we will use \((12)(\cos(-t))\) for the \(x\)-coordinate and \((12)(\sin(-t))\) for the \(y\)-coordinate. Therefore, the parametric equations for this circle are: \(x=12\cos(-t)\) \(y=12\sin(-t)\)

Step 2: Determine the Interval for the Parameter Values

The parameter \(t\) represents the angle in radians, measured from a reference line, as we trace along the circumference of the circle. Since the circle is traced one full revolution, the interval for the parameter values is equivalent to one complete rotation. In terms of radians, one complete rotation is \(2\pi\) radians. Since our reference point starts at \((0,12)\), we need to shift the interval accordingly. As the initial point starts on the positive \(y\)-axis, we start at \(\frac{\pi}{2}\) radians. Therefore, the interval for \(t\) will be: $$ -\frac{\pi}{2} \leq t \leq \frac{3\pi}{2} $$

Step 3: Graph the Circle and Find a Description in Terms of \(x\) and \(y\)

To graph the circle, we need to eliminate \(t\) from the parametric equations: \(x=12\cos(-t)\) \(y=12\sin(-t)\) We can rewrite these equations: \(\cos(-t)=x/12\) \(\sin(-t) = y/12\) Squaring both sides of these equations, we get: \(\cos^2(-t)=\left(\frac{x}{12}\right)^2\) \(\sin^2(-t)=\left(\frac{y}{12}\right)^2\) According to the Pythagorean identity, we have \(\cos^2(-t) + \sin^2(-t) = 1\). Now, we can substitute the values from the equations above: $$ \left(\frac{x}{12}\right)^2 + \left(\frac{y}{12}\right)^2 = 1 $$ Multiplying all terms by \(12^2=144\), we have: $$ x^2 + y^2 = 144 $$ This is the equation in terms of \(x\) and \(y\). It represents a circle centered at the origin with radius \(12\). The graph will be a circle centered at \((0,0)\) with radius \(12\).

Key Concepts

Circle Equation

The circle equation is a mathematical expression that allows us to represent a circle on a coordinate plane. A standard form for the equation of a circle centered at the origin is given by:

• \( x^2 + y^2 = R^2 \)

Here, \( R \) stands for the radius of the circle. If a circle has a different center, say \((h, k)\), then the equation becomes \((x-h)^2 + (y-k)^2 = R^2 \). In essence, the equation shows how far any point \((x, y)\) on the circle is from the center. When squared, this distance must equal the square of the radius. In this exercise, the circle is centered at the origin, making the circle equation simply \( x^2 + y^2 = 144 \) because \( R = 12 \). This equation is derived from the parametric form and encapsulates the essence of a circle's geometry.

Graphing Circles

Graphing circles involves plotting a set of points that form a perfectly round shape. The equation \( x^2 + y^2 = R^2 \) makes it clear that every solution \((x, y)\) lies on the circle. For graphs, it's helpful to:

• Know the radius \( R \) and center \((h, k)\). These define the size and position of the circle.
• Make sure \( R \) distances from the center are maintained in all directions.
• Use symmetry around both axes for accurate graphing.

Since the given problem's circle is centered at the origin with radius 12, we start by plotting the center at \((0, 0)\). Step out 12 units in all directions to mark the initial boundary points on the x and y axes. Connect these dots smoothly and symmetrically to complete the full circle.

Trigonometric Functions

Trigonometric functions are vital in expressing parametric equations for circles. They handle the angular component, where typically:

• \( \cos(t) \) relates to the x-coordinate.
• \( \sin(t) \) relates to the y-coordinate.

For a parametric circle, \( x = R \cos(t) \) and \( y = R \sin(t) \) create the circular path naturally as \( t \) varies. If the circle's path reverses direction (clockwise instead of the typical counterclockwise), the parameter \( t \) is inverted, using \( \cos(-t) \) and \( \sin(-t) \), or equivalently, \( \cos(t) \) and \(-\sin(t) \). Understanding how these functions plot on a unit circle is crucial for depicting any circle parametrically.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry. It states:

• \( \cos^2(t) + \sin^2(t) = 1 \)

This identity lays the groundwork for understanding circular shapes on a grid. When dealing with parametric equations of a circle, we utilize this identity to eliminate the parameter \( t \):1. Start with the parametric forms \( x = 12 \cos(t) \), \( y = 12 \sin(t) \).2. Divide by the radius: \( \cos(t) = \frac{x}{12} \) and \( \sin(t) = \frac{y}{12} \).3. Substitute into the Pythagorean identity: \( \left(\frac{x}{12}\right)^2 + \left(\frac{y}{12}\right)^2 = 1 \).4. Simplify to the circle equation \( x^2 + y^2 = 144 \).As such, the Pythagorean identity not only supports the bridge from trigonometric to Cartesian coordinates but also substantiates the constant radial nature of circles.