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To solve this problem, let's first analyze the given expression:

\[
(\sin \theta + i \cos \theta)^n = \sin n\theta + i \cos n\theta
\]

We need to find the positive integer values of \( n \) (less than 40) that satisfy this equation and then compute the sum of all such \( n \).

### Step-by-Step Solution:

1. **Understanding the Expression:**
   Let:
   \[
   z = \sin \theta + i \cos \theta
   \]
   Then, the equation becomes:
   \[
   z^n = \sin n\theta + i \cos n\theta
   \]

2. **Magnitude and Argument of \( z \):**
   Notice that:
   \[
   |z| = \sqrt{(\sin \theta)^2 + (\cos \theta)^2} = \sqrt{1} = 1
   \]
   Since the magnitude is 1, \( z \) lies on the unit circle in the complex plane.

   The argument of \( z \) is:
   \[
   \arg(z) = \arg(\sin \theta + i \cos \theta) = \theta
   \]

   Therefore, we can write \( z \) in exponential form:
   \[
   z = e^{i\theta}
   \]

   Raising \( z \) to the power \( n \):
   \[
   z^n = (e^{i\theta})^n = e^{in\theta}
   \]

   The right side of the equation is also:
   \[
   \sin n\theta + i \cos n\theta = e^{in\theta}
   \]

   Hence, the equality \( z^n = \sin n\theta + i \cos n\theta \) holds if:
   \[
   e^{in\theta} = e^{i n\theta}
   \]

   Which is always true by Euler's formula. However, we need to find the specific values of \( n \) that satisfy the initial expression for \( n < 40 \).

3. **Finding Values of \( n \):**
   Since \( e^{i n \theta} = e^{i n \theta} \) always holds, we need to examine the periodic properties.

   Knowing that \( e^{i x} = e^{i (x + 2\pi k)} \) for any integer \( k \), we find that the expression repeats every full cycle of \( 2\pi \). For this, we have:

   \[
   n \theta = \theta + 2k\pi
   \]

   Solving for \( n \), we get:
   \[
   n = 1 + \frac{2k\pi}{\theta}
   \]

   Thus, \( n \) takes values based on the increments of \( \frac{2\pi}{\theta} \).

4. **Determine Possible \( n < 40 \):**
   \( n \) can only take values such that it remains a positive integer less than 40. Thus, we need to find all such values and compute their sum.

5. **Sum the Values of \( n \):**

   Based on the above, let's find the sum numerically. 

### Conclusion:

To proceed, I need to know the specific value of \( \theta \). If you provide that, we can calculate the exact values of \( n \) and their sum. 

Would you like to specify \( \theta \) or have further details? Here are some related questions for you:

1. What is the value of \( \theta \) if known?
2. Could you provide any constraints on \( \theta \)?
3. Should we consider particular cases of \( n \)?
4. Do you want to explore properties of trigonometric identities?
5. Are there other contexts or constraints you are curious about regarding this equation?

**Tip:** Understanding the properties of complex numbers on the unit circle can help solve many trigonometric and geometric problems!

For a positive integer n less than 40, what is the sum of n values satisfying (sin θ + i cos θ)^n = sin nθ + i cos nθ?

Explore the problem of finding the sum of positive integer values of n less than 40 that satisfy the equation (sin θ + i cos θ)^n = sin nθ + i cos nθ. This problem involves complex numbers, trigonometry, and Euler's formula, making it suitable for advanced high school or college-level students.

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