1a. Find the modulus and argument of

Here, $x = 4$ and $y = 3$.

Since $x > 0$ and $y > 0$, $\theta$ is in the first quadrant.

The modulus is $\boxed{5}$ and the argument is $\boxed{\arctan\left(\frac{3}{4}\right)}$.

ii) $Z = 1 + i$

Here, $x = 1$ and $y = 1$.

Since $x > 0$ and $y > 0$, $\theta$ is in the first quadrant.

The modulus is $\boxed{\sqrt{2}}$ and the argument is $\boxed{\frac{\pi}{4}}$.

iii) $Z = \sin\theta + i\cos\theta$

Here, $x = \sin\theta$ and $y = \cos\theta$.

Using the identity $\sin^2\theta + \cos^2\theta = 1$:

The standard polar form of a complex number is $r(\cos\phi + i\sin\phi)$.
We have $Z = \sin\theta + i\cos\theta$.
Using the trigonometric identities $\sin\theta = \cos\left(\frac{\pi}{2} - \theta\right)$ and $\cos\theta = \sin\left(\frac{\pi}{2} - \theta\right)$:

Comparing this to the polar form, the argument $\phi$ is $\frac{\pi}{2} - \theta$.
The modulus is $\boxed{1}$ and the argument is $\boxed{\frac{\pi}{2} - \theta}$.

1b. Evaluate

The powers of $i$ follow a cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. This cycle repeats every 4 powers. To evaluate $i^n$, find the remainder of $n$ when divided by 4.

i) $i^{20}$

Since the remainder is 0, $i^{20}$ is equivalent to $i^4$.

The value is $\boxed{1}$.

ii) $Z = -1 + i^2$

We know $i^2 = -1$.

The value is $\boxed{-2}$.

iii) $i^{-4}$

We know $i^4 = 1$.

The value is $\boxed{1}$.

iv) $i^5$

Since the remainder is 1, $i^5$ is equivalent to $i^1$.

The value is $\boxed{i}$.