Question

Beautiful curves Consider the family of curves $$\begin{array}{l}x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right) \\\y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)\end{array}$$ Plot a graph of the curve for the given values of \(a, b,\) and \(c\) with \(0 \leq t \leq 2 \pi\). $$a=7, b=4, c=1$$

Short answer

Question: Plot the curve described by the following parametric equations with \(a=7,\) \(b=4,\) and \(c=1\): $$x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right)$$ $$y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)$$ Compute points (x, y) with t varying from 0 to \(2\pi\), and describe the steps to do so.

Step-by-step solution

Understand the given equations

We are given parametric equations for x and y, as follows: $$x=\left(2+\frac{1}{2} \sin a t\right) \cos \left(t+\frac{\sin b t}{c}\right)$$ $$y=\left(2+\frac{1}{2} \sin a t\right) \sin \left(t+\frac{\sin b t}{c}\right)$$ Here, a, b and c are constants, and we need to plug in the given values of \(a=7,\) \(b=4,\) and \(c=1\).

Replace the values of a, b, and c in the equation

When we replace the values of a, b, and c in the given equations, we get: $$x=\left(2+\frac{1}{2} \sin 7 t\right) \cos \left(t+\sin 4 t\right)$$ $$y=\left(2+\frac{1}{2} \sin 7 t\right) \sin \left(t+\sin 4 t\right)$$

Find the points (x, y) for varying values of t

To find the points (x, y) on the curve, we need to vary t from \(0\) to \(2\pi\). We can compute x(t) and y(t) for various values of t using the above equations.

Plot the graph of the curve

Now that we have computed the points (x, y) for various values of t, we just need to plot these points on a graph. The graph will show the curve for the given values of a, b, and c. Upper bound: \(t=2\pi\). So from \(t=0\) to \(t=2\pi\), compute x(t) and y(t): 1. Choose the values for t in varying increments, e.g., every \(\frac{\pi}{8}\) or \(\frac{\pi}{16}\). 2. For each value of t within the specified range, compute x(t) and y(t) using the equations. 3. Plot the points (x(t), y(t)). You can use graphing software or a calculator to achieve the above steps and visualize the curve. The result is the graph of the curve for the given values of a, b, and c.

Key concepts

Plotting Graphs

Plotting graphs is an essential technique for visualizing mathematical functions and understanding the relationship between variables. It's particularly valuable when dealing with complex functions, such as those involving parametric equations or trigonometric functions.

When plotting the graph of a parametric curve, you typically construct a visual representation on a coordinate system by marking points whose x and y coordinates are determined by the parametric equations for different values of the parameter, such as 't' in our exercise. To plot a graph effectively:

• Identify the range of values for the parameter; in our case, it's from 0 to \(2\pi\).
• Choose a suitable increment for the parameter to ensure enough points are calculated for a smooth curve.
• Calculate the x and y coordinates for each increment using the given equations.
• Plot the points on a grid and connect them to form the curve.

Using graphing software or a calculator with graphing capabilities can significantly simplify the process. It allows for more precise plots, especially for more sophisticated curves, and enables students to explore the effects of changing parameters on the shape of the curve.

Parametric Curves

Parametric curves provide a way to represent a curve by expressing each coordinate (x and y) as a function of a third variable, often called a parameter (in the given exercise, it's 't'). These expressions are known as parametric equations.

The beauty of parametric curves lies in their versatility. Unlike standard function notation where y is a function of x, parametric equations allow us to describe curves where a single function might not be sufficient. This includes loops and other complex shapes not representable by a single function. To understand parametric equations:

• Recognize that each point on the curve is mapped by a value of the parameter 't'.
• Understand that by varying 't', you trace the curve point by point.
• Realize that the path and speed at which the curve is traced depend on how 't' increments.

Improving your understanding of parametric curves can involve plotting different curves with varied parameters to observe changes in their shape and form.

Trigonometric Functions

Trigonometric functions, such as sin and cos, play an integral role in the study of mathematics, especially within the context of parametric equations. These periodic functions describe relationships involving angles, and they are fundamental in defining the behavior of waves and circular motion.

In parametric equations, trigonometric functions help to establish the x and y coordinates in relation to the parameter 't', which often represents an angle in radians. To work with trigonometric functions:

• Understand their periodic nature and how their values repeat over intervals of \(2\pi\).
• Be familiar with key angles and their sine and cosine values.
• Know how to manipulate these functions to change the shape and orientation of parametric curves.

For the given exercise, varying 't' within the trigonometric functions of the parametric equations results in a complex curve that can be visualized by plotting the calculated points. By exploring the influence of trigonometric functions within parametric curves, students can gain insights into how these functions shape the motion described by the equations.