Question

Consider the family of curves \(x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right), y=\left(2+\frac{1}{2} \sin a t\right)\) \(\sin \left(t+\frac{\sin b t}{c}\right) .\) Plot the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\) $$a=6, b=12, c=3$$

Short answer

Question: Given the parametric equations \(x(t) = (2+\frac{a}{b} \sin at) \cos (t+\frac{\sin bt}{c})\) and \(y(t) = (2+\frac{a}{b} \sin at) \sin (t+\frac{\sin bt}{c})\), find the curve for \(a=6\), \(b=12\), and \(c=3\), and plot it for \(t\) ranging from \(0\) to \(2\pi\). Answer: To find and plot the curve for the given values, follow these steps: 1. Substitute the given values of \(a=6\), \(b=12\), and \(c=3\) into the functions for \(x(t)\) and \(y(t)\) to get the specific curve expressions: \(x(t) = \left(2+\frac{1}{2} \sin 6t\right) \cos \left(t+\frac{\sin 12t}{3}\right)\) \(y(t) = \left(2+\frac{1}{2} \sin 6t\right) \sin \left(t+\frac{\sin 12t}{3}\right)\) 2. Create a table of values for \(t\), \(x(t)\), and \(y(t)\) for the given range \(0 \leq t \leq 2 \pi\), using increments of \(\pi/4\) or a smaller increment for a smoother curve. 3. Plot the curve on the \(xy\)-plane using the table of values from Step 2, connecting the points corresponding to the calculated \(x(t)\) and \(y(t)\) values for each \(t\). Include axis labels and a title for the graph.

Step-by-step solution

Substitute the given values of a, b, and c into the functions

First, substitute the given values of \(a=6\), \(b=12\), and \(c=3\) into the functions for \(x(t)\) and \(y(t)\): \(x(t) = \left(2+\frac{1}{2} \sin 6t\right) \cos \left(t+\frac{\sin 12t}{3}\right)\) \(y(t) = \left(2+\frac{1}{2} \sin 6t\right) \sin \left(t+\frac{\sin 12t}{3}\right)\) These are now the specific expressions for \(x(t)\) and \(y(t)\), which describe the curve for the given values of \(a\), \(b\), and \(c\).

Create a table of values for t, x(t), and y(t)

Next, create a table of values for \(t\), \(x(t)\), and \(y(t)\) for the given range \(0 \leq t \leq 2 \pi\). You can use increments of \(\pi/4\) or a smaller increment for a smoother curve: \$ t \quad x(t) \quad y(t) \newline 0 \quad x(0) \quad y(0) \newline \pi/4 \quad x(\pi/4) \quad y(\pi/4) \newline \pi/2 \quad x(\pi/2) \quad y(\pi/2) \newline \dots \$

Plot the curve using the table of values

Using the table of values created in Step 2, plot the curve on the \(xy\)-plane by connecting the points corresponding to the calculated \(x(t)\) and \(y(t)\) values for each \(t\). You can use graphing software or a graphing calculator to assist in plotting the curve. Be sure to include axis labels and a title for the graph to make it clear what curve is being represented.

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate an angle of a right-angled triangle to the ratios of two sides of the triangle. These functions are foundational in understanding waves, cycles, and phenomena that repeat periodically. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each has its own unique graph and characteristics. For parametric curves, these trigonometric functions play a vital role in expressing the \(x\) and \(y\) coordinates.In our exercise, the functions \(x(t)= \left(2+ \frac{1}{2} \sin 6t\right) \cos \left(t+\frac{\sin 12t}{3}\right)\) and \(y(t)= \left(2+ \frac{1}{2} \sin 6t\right) \sin \left(t+\frac{\sin 12t}{3}\right)\) encapsulate both \(\sin\) and \(\cos\) functions. These expressions result in complex wave-like patterns for the plot. Understanding how changes in parameters like \(a, b,\) and \(c\) affect the frequency and behavior is key to mastering parametric equations!

Curve Plotting

Plotting curves from parametric equations involves breaking down the expressions for \(x\) and \(y\) across a range of possible \(t\)-values. This is essentially like telling a story of how a point moves across the coordinate plane as \(t\) progresses.For example, a smooth curve can be plotted by choosing \(t\)-values at regular intervals like \(\pi/4\). By substituting these values into the parametric equations, we obtain specific \(x(t)\) and \(y(t)\) coordinates. These pairs of coordinates give us points to plot on the \(xy\)-plane. Connecting them forms the desired curve, illustrating the parametric relationship.Visual cues on the graph such as arrows can indicate direction, whereas pattern repetitions reflect periodicity. Adding labels and components like a grid makes the analysis more intuitive.

Graphing Calculators

Graphing calculators are powerful tools that enable us to visualize complex functions and relationships, like those in parametric equations. They can immensely simplify the process of plotting curves by quickly computing and plotting large sets of points directly from the equations input.A typical graphing calculator allows input of \(x(t)\) and \(y(t)\) components separately. After inputting the parametric equations, the calculator uses its processing power to generate the curve over your defined range of \(t\)-values. This alleviates the manual burden of computing each point.To utilize a graphing calculator effectively, you must:

• Ensure parameters like \(a, b,\) and \(c\) are entered correctly according to your equation.
• Define the range of \(t\)
• Set an appropriate increment for a smooth curve.
• Use features like tracing or zooming for an in-depth view.

Graphing calculators provide an interactive way to learn and verify parametric curves, making them invaluable for studying trigonometric functions and more.