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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**.

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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.

Find the length of side AB in a right triangle, where angle ABC is 62°, side AC is 12 cm, and side AB is adjacent to the angle.

Learn how to find the length of side AB in a right triangle using the tangent function. Given angle ABC as 62° and side AC as 12 cm, the solution demonstrates the use of trigonometric ratios to solve for the unknown side.

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