Which of the following best describes the shape traced by the curve with parametric equations $x=\sin \left(t\right)$, $y=3\sin \left(2t\right)$, $z=\sin \left(3t\right)$ as $t$ varies from $0$ to $2\pi$?

• A. A circle in the $x-y$ plane
• B. A straight line in space
• C. A helix around the $z$-axis
• D. A three-dimensional Lissajous curve

Answer: (D)

Verified step by step guidance

1. Identify the parametric equations given: \(x = \sin(t)\), \(y = 3\sin(2t)\), and \(z = \sin(3t)\), where \(t\) varies from \$0\( to \)2\pi$.
2. Note that each coordinate is a sine function with different frequencies: \(x\) has frequency 1, \(y\) has frequency 2 (and amplitude 3), and \(z\) has frequency 3.
3. Recall that when parametric equations involve sine or cosine functions with different frequencies in each coordinate, the resulting curve is often a Lissajous curve, which is a complex, periodic figure in multiple dimensions.
4. Since the frequencies in \(x\), \(y\), and \(z\) are different and the amplitudes vary, the curve will not be a simple circle, straight line, or helix, but rather a three-dimensional Lissajous curve.
5. Therefore, the shape traced by the curve is best described as a three-dimensional Lissajous curve, characterized by the interplay of sine waves with different frequencies along each axis.