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 \(0 \leq t \leq 2 \pi\) $$a=18, b=18, c=7$$

Short answer

Answer: To determine the characteristics of the curve, you would need to follow the steps described and actually plot the curve using the given equations and the specified range of \(t\). Generally, you should observe the graph for any symmetry, asymptotes, or other unique features. However, without generating the plot, it is not possible to specifically describe the characteristics of this particular curve.

Step-by-step solution

Replace the parameters with given values

For our specific curve, we will replace the values of \(a\), \(b\), and \(c\) with the given values. We get the parametric equations as follows: \(x=\left(2+\frac{1}{2} \sin 18t\right) \cos \left(t+\frac{\sin 18 t}{7}\right)\) \(y=\left(2+\frac{1}{2} \sin 18t\right) \sin \left(t+\frac{\sin 18 t}{7}\right)\) We will now plot the curve with the given range of \(t\).

Set a range of values for t

Since we are supposed to plot the curve for \(0\leq t\leq 2\pi\), we will create an array or linspace with equally spaced values within this range for our variable \(t\).

Define the functions x(t) and y(t)

Now we will define the functions for x(t) and y(t) based on the parametric equations we have. $$ x(t) = \left(2+\frac{1}{2} \sin 18t\right) \cos \left(t+\frac{\sin 18 t}{7}\right) $$ $$ y(t) = \left(2+\frac{1}{2} \sin 18t\right) \sin \left(t+\frac{\sin 18 t}{7}\right) $$ We will plug in the values of our array or linspace of \(t\) into x(t) and y(t) to generate the corresponding x and y coordinates for the curve.

Plot the curve using x(t) and y(t)

Using a suitable plotting library (such as matplotlib in Python), plot the curve for the given equations and the range of \(t\) values. The curve will be displayed on a coordinate plane with the x and y coordinates generated by our parametric equations.

Interpret the graph

After completing the plot, analyze the graph and describe its characteristics, such as any periods of symmetry, asymptotes, or other unique features. This will help to give a more complete understanding of the curve's behavior for the given parametric equations.

Key concepts

Family of Curves

In mathematics, a family of curves refers to a set of curves that are defined by a particular equation, often depending on one or more parameters. In the given exercise, the family of curves is described by the parametric equations:

• \(x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right)\)
• \(y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)\)

Each curve within the family is determined by different values of the parameters \(a\), \(b\), and \(c\). This indicates that by adjusting these parameters, we can obtain a variety of curves that exhibit different shapes and characteristics.

For instance, the exercise asks to plot the curve specifically for \(a=18\), \(b=18\), and \(c=7\) with \(0 \leq t \leq 2\pi\). This particular setting reveals how these parameters influence the shape of the curve, providing valuable insights into how changes in the parameters affect the graph's appearance.

Trigonometric Functions

Trigonometric functions are fundamental in describing periodic phenomena and play a crucial role in the given parametric equations. Here, both the sine and cosine functions dictate the behavior of the curve. They are employed in the following ways:

• \(x=\left(2+\frac{1}{2} \sin 18t\right) \cos \left(t+\frac{\sin 18 t}{7}\right)\)
• \(y=\left(2+\frac{1}{2} \sin 18t\right) \sin \left(t+\frac{\sin 18 t}{7}\right)\)

The presence of the trigonometric functions \(\sin\) and \(\cos\) induces periodic behavior. This is reflected in how the values of \(x\) and \(y\) oscillate as \(t\) progresses from \(0\) to \(2\pi\). Such periodicity is typical of trigonometric functions due to their inherent wave-like nature.

Additionally, these functions allow the curve to exhibit symmetrical properties. The synchronicity between both the sine and cosine components determines how the curve appears visually, displaying possibly complex patterns.

Coordinate Plane

The coordinate plane serves as the framework on which parametric equations, such as those in this exercise, are graphically interpreted. It is defined by the horizontal axis (x-axis) and vertical axis (y-axis). In the context of the provided parametric equations, plotting them on the coordinate plane allows visualization of the family of curves that these equations generate.

• For \(x\) and \(y\), each set of coordinates \((x, y)\) results from substituting given \(t\)-values into the parametric functions: \(x(t)\) and \(y(t)\).
• Each point on the plot corresponds to a specific value of \(t\) within the range of \(0\) to \(2\pi\).
• This range ensures that the curve forms a complete outline according to the defined frequency and amplitude components from the trigonometric functions.

Through plotting, the coordinate plane offers an immediate visual grasp of how parametric equations manifest as curves, showcasing the behavior and interaction of their trigonometric components.