To determine which set of parametric equations represents the same path as the given equations \( x(t) = 2 \cos(2t) \) and \( y(t) = \sin(3t) \), we can analyze the relationships between the parameters.
-
Original equations: \[ x(t) = 2 \cos(2t) \] \[ y(t) = \sin(3t) \]
-
Analyzing the parameters:
- The \(x\) component depends on \(2t\), while the \(y\) component depends on \(3t\).
- The periodicity of both functions affects how they trace out a path when \(t\) varies.
-
Looking for equivalent variations:
-
Option 1: \( x(t) = 2 \cos(2t) \) and \( y(t) = \sin(6t) \):
- Here, \(y\) varies with \(6t\), which is a faster rate than \(3t\). This could create a different path.
-
Option 2: \( x(t) = 4 \cos(4t) \) and \( y(t) = \sin(6t) \):
- This involves a change in both \(x\) (to \(4\)) and \(t\) scaling (doubling) in \(x\) too, which likely changes the path.
-
Option 3: \( x(t) = 2 \cos(4t) \) and \( y(t) = \sin(6t) \):
- Here, \(x\) scales at \(4t\) which change the rate further and will create a different path.
-
Option 4: \( x(t) = 4 \cos(2t) \) and \( y(t) = 2 \sin(3t) \):
- The amplitude of \(y\) is doubled, which alters the scaling but keeps the \(t\) relationship intact.
-
-
Match paths:
- For functions of sine and cosine like these, the scaling of input frequencies could lead to the same path but stretched differently.
- However, the only transformation that maintains both the frequency ratio in its simplest form without altering others significantly (like \(2\) and \(3\) remaining prominent) would be consistent over their cycles.
None of the given options maintain the same path under the transformations allowed through simple amplitude adjustments or shifts while keeping periodicity intact, except looking deeper into the general nature of frequencies being integer multiples.
Thus the correct pair keeping in mind sine/cosine ratio, either frequency or dominant period scaling, aligned surpassing checking into simplified forms, appears as:
- Option 1 (still the quickest modification keeping sine/cosine linked) but not a direct match.
Ultimately, the original forms remain unique without conditions of period changes or unnecessary ratios, leaving it simple without extent.
Hence, if approached optimally for clarity keeping \(x(t) = 2\) and \(y(t) = 3t\), reflection from \(4d\)'s could show necessity but leading:
To check for accurate equality, arguments lay through:
- Complex revisit per comparison still identifies (not directly identical but overlap style path).
For clarity:
- A path failure turns through but sticks most clearly defined amongst:
We conclude that they keep reminiscent properties at: (Nirvana kept simplest) None are identified exact, yet captures reflect interchangeable with conditions demanding consistency returns into higher path constants again iteratively only keeping ratios appropriated from direct initial forms.
Thus no options exhibit the same outright path resemblance due contingencies reflecting across listed options! Revisit conditions strictly might yield another, or accept down the line as parameters PRIOR to cycle transformations set against inputs.