Problem 45 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

Replacing 2 with -2 flips the deltoid curve, altering its orientation and symmetry.

Step 1: Understanding the Original Parametric Equations

The original equations given are parametric equations: \( x = 2 \cos t + \cos 2t \) and \( y = 2 \sin t - \sin 2t \). These equations describe the motion of a point in the plane as \( t \) varies from \( 0 \) to \( 2 \pi \).

Step 2: Modifying the Coefficients

To solve the problem, we replace each occurrence of \( 2 \) with \(-2\). Therefore, the new equations become \( x = -2 \cos t + \cos 2t \) and \( y = -2 \sin t - \sin 2t \).

Step 3: Simplify the New Equations

Substituting the modifications into the original parametric equations, we simplify them to: \( x = -2 \cos t + \cos 2t \) and \( y = -2 \sin t - \sin 2t \). These modifications reflect a change in the orientation and shape of the curve.

Step 4: Graphing the New Equations

Using a graphing tool or software, plot the new parametric equations against the original range \( 0 \leq t \leq 2\pi \). Compare the graph with the original deltoid to observe changes in shape and orientation.

Key Concepts

Trigonometric Functions

In this exercise, trigonometric functions like cosine and sine are used to define the parametric equations. Parametric equations express the coordinates of points along a curve in terms of a parameter, often denoted as \( t \). These functions are pivotal in describing oscillatory and rotational phenomena, connecting the motion in circular paths to curvilinear paths.

The function \( \cos(t) \) describes the horizontal displacement, while \( \sin(t) \) does the same for vertical displacement. Together, they allow us to plot intricate curves by varying the parameter \( t \) from \( 0 \) to \( 2\pi \). In our equations, \( 2\cos(t) + \cos(2t) \) and \( 2\sin(t) - \sin(2t) \) generate a pattern that repeats as \( t \) completes a full cycle from \( 0 \) to \( 2\pi \). The introduction of coefficients like 2 and -2 modifies the amplitude and direction, contributing to more complex curve shapes.

Graphing Techniques

Graphing parametric equations requires understanding how the parameter \( t \) influences the entire curve in the coordinate system. As \( t \) ranges from \( 0 \) to \( 2\pi \), these equations trace a closed loop known as a deltoid. Using software tools to graph these equations facilitates an exploration of their shape and changes.

With parametric equations, plotting is not as straightforward as simply drawing lines; it requires stepping through the parameter's values to see how the point

• Begins its path at the initial position when \( t = 0 \)
• Ends up back at the starting position by the time \( t = 2\pi \)

To visualize this process, graphing calculators or computer software can plug-in calculated \( x \) and \( y \) points. This gives students a clearer sense of motion and transformation dictated by these functions, especially when coefficients change. Seeing the graphically rendered output aids in understanding how manipulation of parameters affects the vector direction and shape.

Curve Sketching

Curve sketching with parametric equations allows for nuanced insights into altering shapes by changing parameters or coefficients. The exercise illustrates these modifications by replacing \( 2 \) with \(-2 \), leading to an entirely different graph.

The key changes to observe include:

• Orientational Shift: Switching coefficients flips the curve, resulting in new pathways.
• Shape Distortion: The modifications impact the amplitude of the sine and cosine functions, distorting the original deltoid shape.

The task is to explore the visual transformation. It involves predicting the curve's path and logical variations by simply adjusting the formulae. This highlights the intriguing behavior of parametric curves and can be complemented by trying additional values for various coefficients. Students should compare the new shape to the original, analyzing key differences, like how the endpoints connect or how loops may expand or contract. Such competence in sketching these curves builds foundational knowledge for more advanced topics in mathematics.