Question

Use a graphing utility to graph the curve represented by the parametric equations and identify the curve from its equation. $$\begin{array}{rl}& x=2t-2\sin t\\ & y=2-2\cos t\end{array}$$

Short answer

The curve is a curtate cycloid.

Step-by-step solution

Understand the Parametric Equations

The given parametric equations are $x=2t-2\sin t$ and $y=2-2\cos t$. These equations express $x$ and $y$ as functions of the parameter $t$. We need to analyze these functions to understand the shape and behavior of the curve they represent.

Use Trigonometric Identities

Notice that the equations involve trigonometric functions, which suggest a relation to circular or ellipse structures (since $\sin$ and $\cos$ are periodic). Our task is to convert these parametric equations into a more recognizable form, such as the equation of a circle or ellipse.

Eliminate the Parameter t

To find the Cartesian equation, eliminate $t$ by isolating the trigonometric functions. From $x=2t-2\sin t$, express $\sin t$ as $\sin t=t-\frac{x}{2}$. From $y=2-2\cos t$, rearrange to get $\cos t=1-\frac{y}{2}$.

Use Trigonometric Identity ${\sin }^{2}t+{\cos }^{2}t=1$

Substitute $\sin t=t-\frac{x}{2}$ and $\cos t=1-\frac{y}{2}$ into the identity ${\sin }^{2}t+{\cos }^{2}t=1$, obtaining: $${(t-\frac{x}{2})}^2+{(1-\frac{y}{2})}^2=1$$. This equation defines all the points $(x,y)$ that belong to the curve.

Analyze the Equation

The derived equation ${(t-\frac{x}{2})}^2+{(1-\frac{y}{2})}^2=1$ does not immediately simplify to a standard curve. Analyzing deeper by using a graphing utility will help visualize the curve. The graphing utility should show that the curve is an ellipse centered at $(0,2)$ rotated and shifted according to the parametric terms.

Graph the Parametric Equations

Input the equations $x=2t-2\sin t$ and $y=2-2\cos t$ into a graphing utility. Upon graphing, observe the behavior and appearance of the curve, confirming the detailed shape from the exploration in previous steps.

Identify the Curve from the Graph

By analyzing the graphed output, identify the curve. With graphing, observe that the result is a type of trochoid, specifically, a curtate cycloid, based on the equations and parameter interaction.

Key concepts

Trigonometric Functions

Trigonometric functions are all about angles and periodicity. They're central in defining shapes related to circles and waves. In this exercise, we have parametric equations involving $\sin t$ and $\cos t$. These functions describe circular motion or repetitive oscillation patterns, which could hint at curves like circles or ellipses.
Some important properties of these functions include:

• Periodicity: Both $\sin t$ and $\cos t$ complete a full cycle over intervals of $2\pi$, repeating their values, which is why they're always vital in describing circular paths.
• Amplitude: Here, the amplitude is directly impacting how far "out" the curve stretches. With coefficients of 2, they influence the width and height of the curve.
• Pythagorean Identity: One of the key identities involving these functions is ${\sin }^{2}t+{\cos }^{2}t=1$. This identity helps us in eliminating the parameter $t$ and investigating the Cartesian coordinates, transforming our parametric forms into a form closer to typical algebraic expressions.

Overall, these properties aid in translating from parametric to Cartesian forms in understanding the nature of the curve.

Cartesian Equation

A Cartesian equation links $x$ and $y$ variables without parameters like $t$. Converting parametric equations into Cartesian form simplifies understanding and determining the curve's shape.
In this exercise, we start with:

• $x=2t-2\sin t$
• $y=2-2\cos t$

To find a Cartesian equation, we eliminate $t$. Utilizing trigonometric identities, particularly the Pythagorean identity, makes this possible. Rearranging these two equations allows us to express $\sin t$ and $\cos t$ in terms of $x$ and $y$:

• $\sin t=t-\frac{x}{2}$
• $\cos t=1-\frac{y}{2}$

Substituting into the identity ${\sin }^{2}t+{\cos }^{2}t=1$, results in a scalable form $${(t-\frac{x}{2})}^2+{(1-\frac{y}{2})}^2=1$$. This form acts as a bridge between the parametric space and direct geometrical interpretations, assisting in curve exploration.

Curve Identification

Identifying curves is about matching equations to familiar shapes. When given parametric equations, sometimes the resulting expressions don't neatly categorize into standard circles, ellipses, or lines without some game of recognition.
In this task, graphing tools illuminate the path, making the abstract visible with dynamic interpretation. You plot:

• $x=2t-2\sin t$
• $y=2-2\cos t$

Observing the curve obtained by the graph, one may notice the influential role of $\sin t$ and $\cos t$'s periodic nature, recognizing rotational or looping parts typical of trochoid families.
In this scenario, it's discovered that the curve aligns with a "curtate cycloid". This is a special kind of cycloidal curve, often seen in wheels where the tracing point is beneath the rolling circle. Consulting graphing utilities crystallizes predictions, confirming intuition and assisting in analytical deductions.