State the coordinates at E, for which value of θ is the function increasing, and the range of the function?

The point E represents the minimum value of the function. The minimum value of $\cos\theta$ is $-1$, which occurs at $\theta = 180^\circ$.
Substituting this into the function:

The coordinates at E are $\boxed{(180^\circ, -2)}$.

The function $y = \cos\theta - 1$ is a vertical shift of $y = \cos\theta$. The increasing intervals for $\cos\theta$ are from $180^\circ$ to $360^\circ$.
The function is increasing for $\boxed{\theta \in (180^\circ, 360^\circ)}$.

The range of $\cos\theta$ is $[-1, 1]$. For $y = \cos\theta - 1$, we subtract 1 from these values.
The minimum value is $-1 - 1 = -2$.
The maximum value is $1 - 1 = 0$.
The range of the function is $\boxed{[-2, 0]}$.

a.ii) For the function $y = \sin\theta + 1$

The function $y = \sin\theta + 1$ is a vertical shift of $y = \sin\theta$. The increasing intervals for $\sin\theta$ are from $0^\circ$ to $90^\circ$ and from $270^\circ$ to $360^\circ$.
The function is increasing for $\boxed{\theta \in (0^\circ, 90^\circ) \cup (270^\circ, 360^\circ)}$.

The range of $\sin\theta$ is $[-1, 1]$. For $y = \sin\theta + 1$, we add 1 to these values.
The minimum value is $-1 + 1 = 0$.
The maximum value is $1 + 1 = 2$.
The range of the function is $\boxed{[0, 2]}$.

b.i) For the function $y = \sin\theta + 2$

The point E represents the minimum value of the function. The minimum value of $\sin\theta$ is $-1$, which occurs at $\theta = 270^\circ$.
Substituting this into the function:

The coordinates at E are $\boxed{(270^\circ, 1)}$.

The function $y = \sin\theta + 2$ is a vertical shift of $y = \sin\theta$. The increasing intervals for $\sin\theta$ are from $0^\circ$ to $90^\circ$ and from $270^\circ$ to $360^\circ$.
The function is increasing for $\boxed{\theta \in (0^\circ, 90^\circ) \cup (270^\circ, 360^\circ)}$.

The range of $\sin\theta$ is $[-1, 1]$. For $y = \sin\theta + 2$, we add 2 to these values.
The minimum value is $-1 + 2 = 1$.
The maximum value is $1 + 2 = 3$.
The range of the function is $\boxed{[1, 3]}$.

b.ii) For the function $y = \cos\theta$

The standard cosine function $y = \cos\theta$ increases from $180^\circ$ to $360^\circ$.
The function is increasing for $\boxed{\theta \in (180^\circ, 360^\circ)}$.

The range of the standard cosine function $y = \cos\theta$ is $\boxed{[-1, 1]}$.

c.i) For the function $y = 2\cos\theta + \frac{5}{2}$

The point E represents the minimum value of the function. The minimum value of $\cos\theta$ is $-1$, which occurs at $\theta = 180^\circ$.
Substituting this into the function:

The coordinates at E are $\boxed{(180^\circ, \frac{1}{2})}$.

The function $y = 2\cos\theta + \frac{5}{2}$ is a vertical stretch and shift of $y = \cos\theta$. The increasing intervals for $\cos\theta$ are from $180^\circ$ to $360^\circ$.
The function is increasing for $\boxed{\theta \in (180^\circ, 360^\circ)}$.

The range of $\cos\theta$ is $[-1, 1]$.
For $y = 2\cos\theta$, the range is $[2(-1), 2(1)] = [-2, 2]$.
For $y = 2\cos\theta + \frac{5}{2}$, we add $\frac{5}{2}$ to these values.
The minimum value is $-2 + \frac{5}{2} = \frac{-4+5}{2} = \frac{1}{2}$.
The maximum value is $2 + \frac{5}{2} = \frac{4+5}{2} = \frac{9}{2}$.
The range of the function is $\boxed{[\frac{1}{2}, \frac{9}{2}]}$.