Problem 24 - Thomas Calculus

Question

$$x=3 \cos t+\cos 3 t, \quad y=3 \sin t-\sin 3 t ; \quad 0 \leq t \leq 2 \pi$$ What happens if you replace 3 with \(-3\) in the equations for \(x\) and \(y ?\) Graph the new equations and find out.

Step-by-Step Solution

Answer

Replacing 3 with -3 reverses the orientation of the curve.

Step 1: Analyze the Original Parametric Equations

We are given the parametric equations \(x = 3 \cos t + \cos 3t\) and \(y = 3 \sin t - \sin 3t\) for the parameter \(t\) ranging from \(0\) to \(2\pi\). These equations describe a curve in the plane as \(t\) varies.

Step 2: Modify the Parametric Equations

To investigate what happens when 3 is replaced with -3, we change the equations to \(x = -3 \cos t + \cos 3t\) and \(y = -3 \sin t - \sin 3t\). These modified equations will be used to graph the new curve.

Step 3: Graph the Modified Equations

Using the modified equations \(x = -3 \cos t + \cos 3t\) and \(y = -3 \sin t - \sin 3t\), plot the curve by varying \(t\) from \(0\) to \(2\pi\). Graphing software or a suitable calculator can be used for this task.

Step 4: Compare the Graphs

Compare the graph of the modified equations to that of the original equations. Observe changes in orientation, shape, or position of the curve on the coordinate plane to conclude what effect the negative sign has had.

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental when working with parametric equations. They describe the relationship between angles and sides of a right triangle. In the context of parametric equations, these functions allow us to represent coordinates of a point as it moves smoothly along a curve. In our exercise, the functions used are \( \cos t \) and \( \sin t \), as well as \( \cos 3t \) and \( \sin 3t \). Here, \( t \) is a parameter representing an angle, and as \( t \) varies, the value of the trigonometric functions changes accordingly.

By understanding how these functions behave—such as how they repeat every \( 2\pi \) radians, known as their period—we can anticipate how the point \((x,y)\) will trace out specific paths on the coordinate plane. This knowledge helps us interpret and predict the overall shape and behavior of parametric curves.

Basics of Curve Plotting with Parametric Equations

Curve plotting involves visualizing the path described by parametric equations on the coordinate plane. When using parametric equations like those given in the exercise, we calculate the \( x \) and \( y \) coordinates separately for a range of \( t \) values.

• For \( t = 0 \) to \( 2\pi \), inputting values of \( t \) in the equations tells us the precise location of points on the curve.
• The function outputs for \( x \) and \( y \) change smoothly as \( t \) increases, showing how the curve moves and bends in the plane.
• Visual tools and graphing software can be invaluable in observing the entire curve efficiently.

Understanding curve plotting enables us to visualize how the curve is influenced by each parameter in the equations, helping us to see the effects of transformations, such as turning the parameter \( 3 \) into \( -3 \). Through visual observation, key characteristics like direction and shape become much clearer.

Exploring Coordinate Geometry in Parametric Curves

Coordinate geometry provides the framework for analyzing curves produced by parametric equations. Here, each point on a parametric curve is defined by a coordinate pair \((x, y)\) based on the parameter \( t \). This is crucial because it allows us to interpret geometrical properties directly from algebraic expressions.

In the exercise, manipulating the coefficients—changing from 3 to -3—in the equations influences the geometry of the resulting curve. It essentially alters the path of the movement along the plane, switching orientation or even drastically changing the shape.

By comparing the plots with different coefficients, we gain insight into how changes affect the curvature and orientation in coordinate geometry. This understanding is vital in fields such as physics and engineering, where precise control over paths or trajectories is required.