To find the size of angle $A$ in the triangle, we can use the sine rule because we have two sides and an angle that is opposite to one of the sides.

The sine rule states that:

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

Where:

• $a$, $b$, and $c$ are the sides of the triangle opposite angles $A$, $B$, and $C$, respectively.
• $\sin A$, $\sin B$, and $\sin C$ are the sines of the angles opposite those sides.

From the diagram:

• $\angle B = 38^\circ$
• $BC = 9 \, \text{cm}$
• $AB = 11 \, \text{cm}$

We are tasked with finding $\angle A$, so we apply the sine rule as follows:

$\frac{AB}{\sin A} = \frac{BC}{\sin B}$

Substituting the known values:

$\frac{11}{\sin A} = \frac{9}{\sin 38^\circ}$

First, calculate $\sin 38^\circ$:

$\sin 38^\circ \approx 0.6157$

Now, substitute and solve for $\sin A$:

$\frac{11}{\sin A} = \frac{9}{0.6157}$

$\frac{11}{\sin A} \approx 14.615$

$\sin A \approx \frac{11}{14.615} \approx 0.7526$

Now, find $A$ by taking the inverse sine:

$A \approx \sin^{-1}(0.7526) \approx 48.8^\circ$

Thus, $\angle A \approx 48.8^\circ$.

Would you like a detailed breakdown of any specific step or further explanation?

Here are 5 related questions to deepen your understanding:

1. How do you apply the sine rule for non-right triangles in general?
2. What would be the process if you only had two sides and the angle between them?
3. How would you use the cosine rule if the angle wasn't given?
4. What is the difference between the sine and cosine rules, and when should you use each?
5. How do you find the remaining angle $C$ after calculating $A$?

Tip: When using the sine rule, make sure that the angle is opposite the side you are working with to avoid any confusion.