ketch the curve represented by the parametric equations, drawing arrows to indicate the direction of increasing t.
x = 3 cos(𝜃) and y = 2 sin(𝜃) with − 
𝜋
2
 ≤ 𝜃 ≤ 𝜋
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Eliminate the parameter to determine the Cartesian (rectangular) equation of the curve.
  
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 , 
x ≥ 0 or y ≥ 0

The given parametric equations are:

\[
x = 3 \cos(\theta) \quad \text{and} \quad y = 2 \sin(\theta)
\]
with the parameter range:
\[
-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}
\]

### Step 1: Eliminate the parameter to find the Cartesian equation.

To eliminate \(\theta\), we can use trigonometric identities. Since \(x = 3 \cos(\theta)\) and \(y = 2 \sin(\theta)\), let's express \(\cos(\theta)\) and \(\sin(\theta)\) in terms of \(x\) and \(y\):

\[
\cos(\theta) = \frac{x}{3} \quad \text{and} \quad \sin(\theta) = \frac{y}{2}
\]

Now, use the Pythagorean identity:
\[
\cos^2(\theta) + \sin^2(\theta) = 1
\]

Substitute \(\cos(\theta)\) and \(\sin(\theta)\) in this identity:
\[
\left(\frac{x}{3}\right)^2 + \left(\frac{y}{2}\right)^2 = 1
\]

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

This is the equation of an **ellipse** with semi-major axis 3 along the \(x\)-axis and semi-minor axis 2 along the \(y\)-axis.

### Step 2: Determine the direction of increasing \(\theta\).

For \( \theta = -\frac{\pi}{2} \) to \( \theta = \frac{\pi}{2} \):
- At \( \theta = -\frac{\pi}{2} \), \( x = 0 \) and \( y = -2 \) (bottom of the ellipse).
- At \( \theta = 0 \), \( x = 3 \) and \( y = 0 \) (rightmost point of the ellipse).
- At \( \theta = \frac{\pi}{2} \), \( x = 0 \) and \( y = 2 \) (top of the ellipse).

So, the curve starts at the bottom of the ellipse, moves to the right, and then reaches the top. **The direction is counterclockwise**.

### Step 3: Draw the curve.

The curve is an ellipse with axes of 3 and 2, centered at the origin. You should draw arrows in the counterclockwise direction to indicate increasing \(\theta\).

### Step 4: Determine restrictions on \(x\) and \(y\).

Given that \(\theta\) lies between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), the cosine function (which defines \(x\)) is non-negative, meaning \( x \geq 0 \). However, \( y \) can take both positive and negative values because the sine function ranges from \(-1\) to \(1\).

Thus, the region for the curve is restricted to **\( x \geq 0 \)**, but \( y \) can be either positive or negative.

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Would you like more details on any of these steps, or have any other questions?

Here are 5 related questions for further practice:
1. How do you parametrize an ellipse with different center points?
2. What is the effect of changing the range of \(\theta\) on the parametric curve?
3. How do you graph parametric equations using a calculator?
4. What is the physical meaning of parametric curves in motion modeling?
5. How do you calculate the arc length of a parametric curve?

**Tip**: Always check the range of your parameter carefully, as it affects the portion of the curve you are plotting!

Sketch the curve represented by the parametric equations, drawing arrows to indicate the direction of increasing t. x = 3 cos(𝜃) and y = 2 sin(𝜃) with − π/2 ≤ 𝜃 ≤ π/2. Eliminate the parameter to determine the Cartesian (rectangular) equation of the curve.

Learn how to sketch the curve represented by the parametric equations x = 3 cos(𝜃) and y = 2 sin(𝜃) for −π/2 ≤ 𝜃 ≤ π/2. Follow the steps to eliminate the parameter and find the Cartesian equation of the ellipse, understand its direction of increasing t, and analyze the equation in detail. Suitable for high school math students.

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