To express $3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$ in standard form, let's proceed step by step:

1. We are given a complex number in polar form: $r(\cos \theta + i \sin \theta)$ Here, $r = 3$ and $\theta = \frac{3\pi}{2}$.

2. Evaluate the cosine and sine values: $\cos \frac{3\pi}{2} = 0$ $\sin \frac{3\pi}{2} = -1$

3. Substitute these values back into the expression: $3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2}) = 3(0 + i(-1)) = 3(0 - i) = -3i$

So, the standard form of $3(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$ is $-3i$.

Would you like further explanation or details? Here are some related questions to expand on this topic:

1. How do you convert from polar form to rectangular form in general?
2. What is the significance of the angle $\frac{3\pi}{2}$ in the unit circle?
3. How do you find the modulus and argument of a complex number?
4. What is Euler's formula and how does it relate to this problem?
5. How does multiplying a complex number in polar form affect its magnitude and direction?

Tip: Remember that $\cos \frac{3\pi}{2} = 0$ and $\sin \frac{3\pi}{2} = -1$ correspond to the point on the negative imaginary axis in the complex plane.