In a right-angled triangle, the longest side has a length of 300 mm, while the horizontal length is 250 mm. Calculate the vertical length and both included angles.

Let the longest side (hypotenuse) be $c = 300$ mm and the horizontal length be $a = 250$ mm. Let the vertical length be $b$.
The Pythagorean theorem states $a^2 + b^2 = c^2$.

$$(250\, \text{mm})^2 + b^2 = (300\, \text{mm})^2$$ $$62500\, \text{mm}^2 + b^2 = 90000\, \text{mm}^2$$ $$b^2 = 90000\, \text{mm}^2 - 62500\, \text{mm}^2$$ $$b^2 = 27500\, \text{mm}^2$$ $$b = \sqrt{27500}\, \text{mm}$$ $$b \approx 165.83\, \text{mm}$$

The vertical length is approximately 165.83 mm.

Let $\theta_1$ be the angle opposite the vertical side ($b$) and adjacent to the horizontal side ($a$).

$$\cos(\theta_1) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{250\, \text{mm}}{300\, \text{mm}}$$ $$\cos(\theta_1) = \frac{25}{30} = \frac{5}{6}$$ $$\theta_1 = \arccos\left(\frac{5}{6}\right)$$ $$\theta_1 \approx 33.56^\circ$$

Let $\theta_2$ be the angle opposite the horizontal side ($a$) and adjacent to the vertical side ($b$).

$$\sin(\theta_2) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{250\, \text{mm}}{300\, \text{mm}}$$ $$\sin(\theta_2) = \frac{5}{6}$$ $$\theta_2 = \arcsin\left(\frac{5}{6}\right)$$ $$\theta_2 \approx 56.44^\circ$$

Alternatively, since the sum of angles in a triangle is $180^\circ$ and one angle is $90^\circ$:

$$\theta_2 = 180^\circ - 90^\circ - \theta_1 = 90^\circ - 33.56^\circ \approx 56.44^\circ$$

The vertical length is approximately 165.83 mm, and the included angles are approximately $33.56^\circ$ and $56.44^\circ$.

3. Work out the longest side of a right-angled triangle if the shorter sides are 50 mm and 220 mm.

Let the shorter sides be $a = 50$ mm and $b = 220$ mm. Let the longest side (hypotenuse) be $c$.

$$c^2 = a^2 + b^2$$ $$c^2 = (50\, \text{mm})^2 + (220\, \text{mm})^2$$ $$c^2 = 2500\, \text{mm}^2 + 48400\, \text{mm}^2$$ $$c^2 = 50900\, \text{mm}^2$$ $$c = \sqrt{50900}\, \text{mm}$$ $$c \approx 225.61\, \text{mm}$$

The longest side is approximately $\boxed{225.61\, \text{mm}}$.

4. Work out the shorter side of a right-angled triangle if the longest side is 120 mm and the other shorter side is 90 mm.

Let the longest side (hypotenuse) be $c = 120$ mm and one shorter side be $a = 90$ mm. Let the other shorter side be $b$.

$$a^2 + b^2 = c^2$$ $$(90\, \text{mm})^2 + b^2 = (120\, \text{mm})^2$$ $$8100\, \text{mm}^2 + b^2 = 14400\, \text{mm}^2$$ $$b^2 = 14400\, \text{mm}^2 - 8100\, \text{mm}^2$$ $$b^2 = 6300\, \text{mm}^2$$ $$b = \sqrt{6300}\, \text{mm}$$ $$b \approx 79.37\, \text{mm}$$

The shorter side is approximately $\boxed{79.37\, \text{mm}}$.

5. Use your calculator to work out the ratios for these angles:

5.1 sin 20°

$$\sin(20^\circ) \approx \boxed{0.3420}$$

5.2 cos 30°

$$\cos(30^\circ) \approx \boxed{0.8660}$$

5.3 tan 45°

$$\tan(45^\circ) = \boxed{1}$$

6. Use your calculator to work out the angles for these ratios:

6.1 sin x = 0.8337

$$x = \arcsin(0.8337)$$ $$x \approx \boxed{56.48^\circ}$$

6.2 cos x = 8/14

$$\cos x = \frac{8}{14} = \frac{4}{7}$$ $$x = \arccos\left(\frac{4}{7}\right)$$ $$x \approx \boxed{55.15^\circ}$$

6.3 tan x = 32

$$x = \arctan(32)$$ $$x \approx \boxed{88.21^\circ}$$