In a right-angled triangle, if one angle is 30 degrees and the adjacent side is 600, what is the length of the opposite side H? Use sin 30 = 0.5.

We have a right-angled triangle with an angle of $30^\circ$. The side opposite this angle is $H$, and the side adjacent to this angle is 600. We are given $\sin 30^\circ = 0.5$.
To relate the opposite and adjacent sides, we use the tangent function:

$$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$ $$\tan 30^\circ = \frac{H}{600}$$

We know that $\tan \theta = \frac{\sin \theta}{\cos \theta}$. First, we find $\cos 30^\circ$ using the identity $\sin^2 \theta + \cos^2 \theta = 1$:

$$\cos 30^\circ = \sqrt{1 - \sin^2 30^\circ}$$ $$\cos 30^\circ = \sqrt{1 - (0.5)^2}$$ $$\cos 30^\circ = \sqrt{1 - 0.25}$$ $$\cos 30^\circ = \sqrt{0.75} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$$

Now, calculate $\tan 30^\circ$:

$$\tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{0.5}{\frac{\sqrt{3}}{2}} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}}$$

Substitute the value of $\tan 30^\circ$ into the equation from Step 1:

$$\frac{1}{\sqrt{3}} = \frac{H}{600}$$ $$H = \frac{600}{\sqrt{3}}$$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{3}$:

$$H = \frac{600\sqrt{3}}{3}$$ $$H = 200\sqrt{3}$$

Using the approximate value $\sqrt{3} \approx 1.732$:

$$H \approx 200 \times 1.732$$ $$H \approx 346.4$$

The value of $H$ for the first triangle is \boxed{\text{200\sqrt{3} or approximately 346.4}}

For the second triangle:

We have a right-angled triangle with an angle of $60^\circ$. The side opposite this angle is $H$, and the hypotenuse is $30 \text{ cm}$. We are given $\sin 60^\circ = 0.9$.
To relate the opposite side and the hypotenuse, we use the sine function:

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\sin 60^\circ = \frac{H}{30 \text{ cm}}$$

Substitute the given value of $\sin 60^\circ = 0.9$ into the equation from Step 1:

$$0.9 = \frac{H}{30 \text{ cm}}$$ $$H = 0.9 \times 30 \text{ cm}$$ $$H = 27 \text{ cm}$$

The value of $H$ for the second triangle is $\boxed{\text{27 cm}}$