Question

Question
Identify the parametric equations that represent the same path as the following parametric equations.

x left parenthesis t right parenthesis equals 2 cosine 2t y left parenthesis t right parenthesis equals sine 3t

(1 point)
Responses

x left parenthesis t right parenthesis equals 2 cosine 2t y left parenthesis t right parenthesis equals sine 6t
Image with alt text: x left parenthesis t right parenthesis equals 2 cosine 2t y left parenthesis t right parenthesis equals sine 6t

x left parenthesis t right parenthesis equals 4 cosine 4t y left parenthesis t right parenthesis equals sine 6t
Image with alt text: x left parenthesis t right parenthesis equals 4 cosine 4t y left parenthesis t right parenthesis equals sine 6t

x left parenthesis t right parenthesis equals 2 cosine 4t y left parenthesis t right parenthesis equals sine 6t
Image with alt text: x left parenthesis t right parenthesis equals 2 cosine 4t y left parenthesis t right parenthesis equals sine 6t

x left parenthesis t right parenthesis equals 4 cosine 2t y left parenthesis t right parenthesis equals 2 sine 3t

Answer 1

To determine which set of parametric equations represents the same path as the original equations \( x(t) = 2 \cos(2t) \) and \( y(t) = \sin(3t) \), we need to examine if we can manipulate one of the proposed equations to match the original equations or if they can be derived from each other through appropriate transformations like scaling or changing the parameter.

1. **Given Equations**:
- \( x(t) = 2 \cos(2t) \)
- \( y(t) = \sin(3t) \)

### Analyze Proposed Parametric Equations:

**1.** \( x(t) = 2 \cos(2t) \), \( y(t) = \sin(6t) \)

- \( y(t) = \sin(6t) \) has a different frequency (6 instead of 3), so it does not represent the same path.

**2.** \( x(t) = 4 \cos(4t) \), \( y(t) = \sin(6t) \)

- Again, both the amplitude of \( x(t) \) and the frequency of \( y(t) \) have changed, so it does not represent the same path.

**3.** \( x(t) = 2 \cos(4t) \), \( y(t) = \sin(6t) \)

- Similar to the other options, the frequency of the cosine equation is doubled and the sine equation has a frequency of 6, so it also does not match.

**4.** \( x(t) = 4 \cos(2t) \), \( y(t) = 2 \sin(3t) \)

- Now, we can check if this can be simplified. The \( x(t) \) function has its amplitude scaled by a factor of 2, and the \( y(t) \) function is also scaled by 2. If we fix the scaling, we can reparameterize or multiply by constants.

This suggests that it could be equivalent, as scaling either both equations might yield the same path.

### Finding Parameter Relations:

To check pairs:
- For the fourth option:
- Define a new parameter \( s = 2t \):
\[
x(s) = 4 \cos(s) \text{ (where } s = 2t\text{) } = 2 \cos(2t)
\]
\[
y(s) = 2 \sin(3t) = 2 \cdot \sin(\frac{3s}{2})
\]

Scaling preserves the path in parametric equations, thus when reparameterizing, they can coincide back to the original path.

### Conclusion:

The option that maintains the form and allows for scaling while remaining a shape described by the original functions is:

**Final Answer**: \( x(t) = 4 \cos(2t) \) and \( y(t) = 2 \sin(3t) \) (the fourth option).

This choice accurately matches the original parametric equations when taken transformations into account.