The given triangle has two angles, ∠A = 42° and ∠B = 75°, and side $AB = 22 \, \text{cm}$. We are tasked with finding the unknown side $CB$, which is opposite the given angle $A$.

We can solve this problem using the Law of Sines, which states:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

Step 1: Determine ∠C

The sum of the angles in a triangle is always 180°. Therefore, we can calculate ∠C as:

$\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 42^\circ - 75^\circ = 63^\circ$

Step 2: Use the Law of Sines

Now, using the Law of Sines:

$\frac{CB}{\sin(A)} = \frac{AB}{\sin(C)}$

Plugging in the known values:

$\frac{CB}{\sin(42^\circ)} = \frac{22}{\sin(63^\circ)}$

Step 3: Solve for $CB$

Let me compute the result for you.The length of side $CB$ is approximately $16.5 \, \text{cm}$, rounded to the nearest tenth.

Would you like further details or have any additional questions?

Here are 5 related questions you might explore:

1. How can you use the Law of Cosines to solve for a side in a triangle?
2. What other methods can you use to solve triangles when given different combinations of sides and angles?
3. How would the Law of Sines apply if you knew two sides and an angle opposite one of them?
4. Can the Law of Sines be used in non-right triangles, and if so, how?
5. What are the possible limitations of using the Law of Sines in certain triangle configurations?

Tip: Always make sure to calculate the sum of angles in a triangle to ensure your angle values are accurate!