To eliminate $t$ and write a rectangular equation in terms of $x$ and $y$, we will use trigonometric identities.

You are given:

$x(t) = 3\sin(t)$ $y(t) = 2\cos(t)$

Step 1: Express $\sin(t)$ and $\cos(t)$ in terms of $x$ and $y$

From the equation for $x(t)$:

$\sin(t) = \frac{x}{3}$

From the equation for $y(t)$:

$\cos(t) = \frac{y}{2}$

Step 2: Use the Pythagorean identity

The trigonometric identity $\sin^2(t) + \cos^2(t) = 1$ allows us to eliminate $t$:

$\left( \frac{x}{3} \right)^2 + \left( \frac{y}{2} \right)^2 = 1$

Step 3: Simplify

$\frac{x^2}{9} + \frac{y^2}{4} = 1$

This is the equation of an ellipse.

Final Answer:

$\frac{x^2}{9} + \frac{y^2}{4} = 1$

This is the rectangular equation in terms of $x$ and $y$, where $t$ has been eliminated.

Would you like more details or have any questions?

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Here are 5 related questions you might want to explore:

1. How do you derive the standard form of an ellipse from parametric equations?
2. What are the geometric interpretations of the constants in the equation of an ellipse?
3. How can you convert parametric equations to rectangular equations in general?
4. Can you graph this ellipse based on the rectangular equation?
5. How do the limits of $t$ (from 0 to $2\pi$) affect the graph of this equation?

Tip: The coefficients of $x^2$ and $y^2$ in an ellipse equation represent the squares of the semi-major and semi-minor axes lengths!