We are given a right triangle $\triangle ABC$, where angle $\angle ABC = 62^\circ$, side $AC = 12 \, \text{cm}$, and we need to find the length of side $AB$.

Steps to solve:

1. Labeling the triangle:

   • $\angle A = 90^\circ$ (since it's a right-angled triangle).
   • $AB$ is adjacent to angle $B = 62^\circ$.
   • $AC = 12 \, \text{cm}$ is the opposite side to angle $B = 62^\circ$.

2. Trigonometric function: To find $AB$, we can use the tangent function, which relates the opposite side and the adjacent side in a right triangle.

   $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ Here, $\theta = 62^\circ$, the opposite side is $AC = 12 \, \text{cm}$, and the adjacent side is $AB$.

3. Applying the formula: $\tan(62^\circ) = \frac{12}{AB}$ Rearranging to solve for $AB$: $AB = \frac{12}{\tan(62^\circ)}$

4. Calculating: Using a calculator: $\tan(62^\circ) \approx 1.8807$ Therefore: $AB = \frac{12}{1.8807} \approx 6.4 \, \text{cm}$

So, the length of side $AB$ is approximately 6.4 cm.

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How would the problem change if you were asked to find side $BC$ instead?
2. Can you use the sine or cosine functions to find side $AB$ in this triangle?
3. How would the answer change if the given angle was 45° instead of 62°?
4. What is the area of triangle $ABC$ based on the given sides?
5. How can you check your answer using the Pythagorean theorem?

Tip: When solving triangles, always choose the trigonometric function that relates the known and unknown sides most directly.