Problem 23 - Thomas Calculus

Question

$$x=2 \cos t+\cos 2 t, \quad y=2 \sin t-\sin 2 t ; \quad 0 \leq t \leq 2 \pi$$ 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 mirrors and scales the parametric curve.

Step 1: Substitute -2 into the equations

Substitute the value of -2 instead of 2 in the given parametric equations.The equations become: \[ x = -2 \cos t + \cos 2t, \quad y = -2 \sin t - \sin 2t \]

Step 2: Understand the new equation format

The transformed parametric equations represent new functions for the curve. We expect these changes will alter the curve's orientation and location because the scalar factor affects amplitude, and thus, the range of the motion in both x and y directions.

Step 3: Determine the impact on periodicity

Since the trigonometric functions (sine and cosine) maintain the same periodic nature, the parametrization does not change period, remaining at \( 0 \leq t \leq 2 \pi \). The negative factor affects the direction of the curve.

Step 4: Sketch the graph based on new equations

Plot \( x = -2 \cos t + \cos 2t \) and \( y = -2 \sin t - \sin 2t \). The graph should appear as a rotated and scaled version of the original (with 2 instead of -2), typically mirrored across an axis or diagonal depending on symmetry.

Key Concepts

Trigonometric Functions

In parametric equations, trigonometric functions like sine and cosine play a crucial role in defining the curve. These functions cycle through a range of values as the parameter, typically denoted as \( t \), changes. This range will repeat in a periodic manner, usually every \(2\pi\) for both sine and cosine.

• **Sine (\( \sin t \)) and Cosine (\( \cos t \)):** These functions oscillate between -1 and 1, and their unique combination in parametric forms can model complex motion, such as rotations or waves.
• **Compound Trigonometric Formulas:** Functions like \( \cos 2t \) or \( \sin 2t \) amplify the frequency of oscillation, creating curves with more loops or waves over the same period.

To manipulate these trigonometric functions in parametric equations directly affects how your graph behaves. For example, changing a coefficient in front of \( \cos t \) or \( \sin t \) from positive to negative not only scales but flips the function's graph. This effect is particularly important in our equation's transformations, where we've observed a reflection across an axis due to these changes.

Graphing

Graphing parametric equations can be a fascinating journey! The key is plotting points as the parameter \( t \) changes, revealing the path traced by the equations over time. The challenge is not just plotting, but predicting the graph's behavior from its equations.When altering parametric equations, you're not just affecting the path’s coordinates but also its visual representation.

• **Reversal and Scaling:** Replacing the coefficient '2' with '-2' scales and reverses the amplitude. This causes shifts and reflections in the curve that are essential to note. Look for changes in direction and size when drawing curves.
• **Symmetry Consideration:** Symmetry often helps us predict graph behavior. Functions with cosine (even) and sine (odd) terms may reflect over certain axes when signs are changed. Consider, for instance, how reversing signs in our exercise creates a mirror effect for the original curve.

By plotting both equations, \( x = -2 \cos t + \cos 2t \) and \( y = -2 \sin t - \sin 2t \), we see transformations like rotations and flips, offering a glimpse into the effect of negative coefficients in parametric equations.

Curve Orientation

Curve orientation in parametric equations defines the path taken over time by a set of parametric equations. Understanding orientation helps determine where the curve starts and ends, and how it moves in relation to geometric axes. Changes in sign or coefficient can impact orientation:

• **Directional Influence:** A negative sign in the parametric coefficients can switch the direction of motion along the curve. In our revised equations, the traversal occurs in the opposite direction due to negative multipliers.
• **Curve Reflection:** With parametric curves, negative scaling results in reflecting the curve across an origin or axis. This effectively shifts the orientation, making the path retrospectively align with its mirrored counterpart.

Understanding these changes in curve orientation is key when analyzing parametric curves graphically. These subtle shifts not only aid in visual comprehension but also enhance problem-solving skills by predicting the curve’s behavior on a cartesian plane.