To solve for angle $x$ in the given right-angled triangle using trigonometry, we can use the tangent function since we are given the opposite side (3 cm) and the adjacent side (8 cm) to angle $x$.

The formula for the tangent of an angle in a right-angled triangle is:

$\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{8}$

Now, to find $x$, we take the inverse tangent ($\tan^{-1}$) of the ratio:

$x = \tan^{-1} \left( \frac{3}{8} \right)$

Let me calculate that for you.The size of angle $x$ is approximately $20.56^\circ$.

Would you like more details or have any questions about the solution?

Here are 5 related questions for further practice:

1. What is the sine of angle $x$ in this triangle?
2. How would you find the hypotenuse of the triangle?
3. Can you calculate the area of the triangle?
4. What would the angle $x$ be if the opposite side was 5 cm instead of 3 cm?
5. How would you use the cosine function to find another angle in the triangle?

Tip: When dealing with right-angled triangles, always identify the sides relative to the angle in question (opposite, adjacent, and hypotenuse) before choosing the correct trigonometric ratio.