Question

Eliminate the parameter to find a description of the following circles or circular arcs in terms of \(x\) and \(y .\) Give the center and radius, and indicate the positive orientation. $$x=3 \cos t, y=3 \sin t ; \pi \leq t \leq 2 \pi$$

Short answer

Question: Determine the equation of the curve, its center, radius, and orientation based on the parametric equations \(x = 3 \cos t\) and \(y = 3 \sin t\), where \(\pi \leq t \leq 2\pi\). Answer: The equation of the curve is \(x^2 + y^2 = 9\), with center at (0,0), radius 3, and a counterclockwise positive orientation.

Step-by-step solution

Observe any relationship between given x and y

Since the given functions are: $$x = 3 \cos t$$ $$y = 3 \sin t$$ Observe that squaring both equations and summing them will give: $$x^2 + y^2 = 9(\cos^2 t + \sin^2 t)$$ Since the Pythagorean identity is \(\cos^2 t + \sin^2 t = 1\), we have: $$x^2 + y^2 = 9(1)$$

Eliminate the parameter t

Now we have eliminated the parameter \(t\), and the equation of the circle or the circular arc is given by: $$x^2 + y^2 = 9$$

Find the center and radius

The given equation can be written as: $$(x-0)^2 + (y-0)^2 = 3^2$$ Which is a circle with center \((0,0)\) and radius \(r = 3\).

Determine the orientation

The given range for parameter \(t\) is: $$ \pi \leq t \leq 2\pi $$ Since this range covers the second and third quadrants, we can say that the curve moves counterclockwise.

Summarize the information

The given curve can be described as: $$x^2 + y^2 = 9$$ With center \((0,0)\), radius \(3\), and a counterclockwise positive orientation from \(\pi \leq t \leq 2\pi\).

Key concepts

Circular Arc Equations

Circular arc equations represent the mathematical description of a portion of a circle. In the context of parametric equations, such as
\(x = 3 \cos t\) and \(y = 3 \sin t\), the parameters \(t\) vary over a specific range to describe a circular arc. This range is critical because it distinguishes a full circle from an arc. The unit circle, which these parametric equations are based on, is a circle with a radius of one centered at the origin.

With parameters, the position along the arc can be identified for any value of \(t\), allowing for the representation of circular motion or the tracing of a circular path. The equations provided in the exercise represent the x and y coordinates of a point on the circle of radius 3, at an angle \(t\) from the positive x-axis.

Eliminating the Parameter

The process of eliminating the parameter in parametric equations involves converting these equations into a single equation that doesn’t involve the parameter. This is often done using algebraic manipulation, trigonometric identities, or geometric insights into the problem.

In our exercise, eliminating the parameter \(t\) was achieved by recognizing that \(x\) and \(y\) are related to the trigonometric functions cosine and sine, respectively. Squaring both parametric equations and adding them utilized the Pythagorean identity to transform the equations into a standard form that represents a circle or a circular arc in Cartesian coordinates. This form, \(x^2 + y^2 = r^2\), immediately suggests the shape and size of the circle without needing to reference the parameter \(t\).

Pythagorean Identity in Trigonometry

The Pythagorean identity in trigonometry is a fundamental relation between the sine and cosine of an angle. It states that for any angle \(t\):
\[\cos^2 t + \sin^2 t = 1\]
This identity is derived from the Pythagorean theorem applied to the unit circle, where the hypotenuse is always 1, and the legs are the cosine and sine of the angle, representing the x and y coordinates, respectively.

In our exercise, the Pythagorean identity was used to transition from parametric equations to a standard circle equation by squaring and summing the values of \(x\) and \(y\) as given by the cosine and sine functions. This identity is crucial for understanding the inherent circular nature of the equations given in parametric form.

Finding the Center and Radius of a Circle

To find the center and radius of a circle given in the standard equation form \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius, you need to identify the values associated with \(h\), \(k\), and \(r\).

In the default form, \(h\) and \(k\) will correspond to the numbers that the variables \(x\) and \(y\) would have to be to make the equation zero. The value of \(r\) is found by taking the square root of the constant term on the right side of the equation. When the equation does not specifically have \(h\) and \(k\), such as \(x^2 + y^2 = 9\), this indicates that the circle is centered at the origin, and \(r\) is the square root of 9, which is 3. Identifying the center and radius is not only valuable for visualizing the circle but also for understanding its geometric properties, such as area and circumference.