Given the following parametric equations, eliminate the parameter, t*t*​, and rewrite the parametric equations as a Cartesian equation. 

x(t)=−3sin⁡(t)y(t)=2cos⁡(t)*x*(*t*)*y*(*t*)​=−3sin(*t*)=2cos(*t*)​​where 0≤t≤2π0≤*t*≤2*π*​​. 




To eliminate the parameter \( t \) from the given parametric equations and rewrite them as a Cartesian equation, we can use trigonometric identities. The given parametric equations are:

\[ x(t) = -3 \sin(t) \]
\[ y(t) = 2 \cos(t) \]

We start by isolating the trigonometric functions in terms of \( x \) and \( y \):

1. From \( x(t) = -3 \sin(t) \):
   \[ \sin(t) = -\frac{x}{3} \]

2. From \( y(t) = 2 \cos(t) \):
   \[ \cos(t) = \frac{y}{2} \]

Next, we use the Pythagorean identity for sine and cosine, which states:
\[ \sin^2(t) + \cos^2(t) = 1 \]

Substituting the expressions for \(\sin(t)\) and \(\cos(t)\) into this identity, we get:
\[ \left(-\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1 \]

Simplifying this, we obtain:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

This is the equation of an ellipse in Cartesian coordinates. Thus, the parametric equations \( x(t) = -3 \sin(t) \) and \( y(t) = 2 \cos(t) \) can be rewritten as the Cartesian equation:
\[ \frac{x^2}{9} + \frac{y^2}{4} = 1 \]

If you have any questions or need further details, please let me know.

Here are 5 related questions for further practice:

1. What is the parametric form of the ellipse \(\frac{x^2}{16} + \frac{y^2}{25} = 1\)?
2. How would you convert the parametric equations \( x(t) = 5 \cos(t) \) and \( y(t) = 7 \sin(t) \) to Cartesian form?
3. What are the major and minor axes of the ellipse described by \(\frac{x^2}{36} + \frac{y^2}{49} = 1\)?
4. If the parametric equations are given by \( x(t) = 4 \cos(t) \) and \( y(t) = 3 \sin(t) \), what is the range of \( t \) to complete one full cycle of the ellipse?
5. How can you determine the orientation of an ellipse given its parametric equations?

**Tip:** When converting parametric equations to Cartesian form, always look for trigonometric identities or algebraic relationships that can help eliminate the parameter.

Given the following parametric equations, eliminate the parameter, t, and rewrite the parametric equations as a Cartesian equation. x(t) = −3sin(t), y(t) = 2cos(t), where 0 ≤ t ≤ 2π.

Learn how to eliminate the parameter from the parametric equations x(t) = −3sin(t) and y(t) = 2cos(t), and convert them to the Cartesian form of an ellipse. This step-by-step solution involves trigonometric identities and results in the equation of an ellipse. Suitable for Grades 10-12 students.

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