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!