https://r2.erweima.ai/mathbot/image/1723854483802qr2cmjue.jpg

Let's analyze and solve the problem given in the image.

### Problem Summary:
The curve is described by the parametric equations:
\[
x = 1 + \sin t, \quad y = 2 \cos t, \quad 0 \leq t < 2\pi
\]

There are two parts to this problem:
1. **Part (a)**: Find the coordinates of the point on the curve when \( t = \frac{\pi}{2} \).
2. **Part (b)**: Determine the value of \( t \) corresponding to the point on the curve with coordinates \(\left(\frac{3}{2}, -\sqrt{3}\right)\).

### Solution:

#### Part (a):
Substitute \( t = \frac{\pi}{2} \) into the parametric equations.

- \( x = 1 + \sin\left(\frac{\pi}{2}\right) = 1 + 1 = 2 \)
- \( y = 2 \cos\left(\frac{\pi}{2}\right) = 2 \times 0 = 0 \)

So, the coordinates are:
\[
(2, 0)
\]

#### Part (b):
We need to find the value of \( t \) such that \( x = \frac{3}{2} \) and \( y = -\sqrt{3} \).

1. **For \( x = \frac{3}{2} \):**
   \[
   1 + \sin t = \frac{3}{2}
   \]
   \[
   \sin t = \frac{3}{2} - 1 = \frac{1}{2}
   \]

   \( \sin t = \frac{1}{2} \) implies \( t = \frac{\pi}{6} \) or \( t = \frac{5\pi}{6} \) (considering \( 0 \leq t < 2\pi \)).

2. **For \( y = -\sqrt{3} \):**
   \[
   2 \cos t = -\sqrt{3}
   \]
   \[
   \cos t = -\frac{\sqrt{3}}{2}
   \]

   \( \cos t = -\frac{\sqrt{3}}{2} \) implies \( t = \frac{5\pi}{6} \) or \( t = \frac{7\pi}{6} \) (considering \( 0 \leq t < 2\pi \)).

The value of \( t \) that satisfies both conditions is \( t = \frac{5\pi}{6} \).

So, the answer is \( t = \frac{5\pi}{6} \).

---

Would you like further details or explanations on any step? Here are some related questions for further practice:

1. What are parametric equations and how do they describe curves?
2. How do you determine the range of \( t \) for a given parametric equation?
3. What is the significance of \( \sin \) and \( \cos \) functions in parametric equations?
4. How would you find the derivative of a curve given in parametric form?
5. How can you find the length of a curve defined by parametric equations?
6. What are the possible values of \( t \) when \( \sin t = -\frac{1}{2} \)?
7. How would you determine the area under a parametric curve?
8. How can you determine if a curve intersects itself based on its parametric equations?

**Tip:** When dealing with trigonometric functions in parametric equations, always consider all possible angles within the given interval that satisfy the equation.

Explore the solution to a parametric equations problem involving the analysis of a curve described by x = 1 + sin(t) and y = 2 cos(t). Learn how to find specific points on the curve and solve for the parameter t given specific coordinates.

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