If `z= 5e^(itheta),` then `|e^(iz)|` is equal to

Options

A. `e^(-5 cos theta)`
B. `e^(5 cos theta)`
C. `e^(5 sin theta)`
D. `e^(-5 sin theta)`

Answer

To solve the problem, we need to find the value of \(|e^{iz}|\) where \(z = 5e^{i\theta}\).

### Step-by-Step Solution:

1. **Express \(z\)**:
\[ z = 5e^{i\theta} \]

2. **Calculate \(iz\)**:
\[ iz = i(5e^{i\theta}) = 5ie^{i\theta} \]

3. **Rewrite \(e^{iz}\)**:
\[ e^{iz} = e^{5ie^{i\theta}} = e^{5i(\cos\theta + i\sin\theta)} = e^{5i\cos\theta - 5\sin\theta} \]

4. **Separate the real and imaginary parts**:
\[ e^{iz} = e^{-5\sin\theta} \cdot e^{5i\cos\theta} \]

5. **Find the modulus**:
The modulus of a product is the product of the moduli:
\[ |e^{iz}| = |e^{-5\sin\theta}| \cdot |e^{5i\cos\theta}| \]

6. **Calculate each modulus**:
- For \(e^{-5\sin\theta}\):
\[ |e^{-5\sin\theta}| = e^{-5\sin\theta} \]
- For \(e^{5i\cos\theta}\):
\[ |e^{5i\cos\theta}| = 1 \quad (\text{since } |e^{i\theta}| = 1) \]

7. **Combine the results**:
\[ |e^{iz}| = e^{-5\sin\theta} \cdot 1 = e^{-5\sin\theta} \]

### Final Answer:
\[ |e^{iz}| = e^{-5\sin\theta} \]

---