To find the length of side $a$ in the given triangle with angles $\angle A = 90^\circ$, $\angle B = 47^\circ$, and side $c$ with a length of 7.7 cm, we can use trigonometric relationships in a right triangle. Since $\angle A$ is $90^\circ$, the given triangle is a right triangle, and we can use the following relationships:

1. Sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

2. Cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Here, the hypotenuse $c$ is 7.7 cm. We need to find the length of side $a$, which is opposite to angle $B$.

Using the sine function: $\sin(B) = \frac{a}{c}$ Solving for $a$: $a = c \cdot \sin(B)$

Given: $\sin(47^\circ) \approx 0.7314$

Therefore: $a = 7.7 \cdot 0.7314 \approx 5.63$

To the nearest tenth of a unit, the length of side $a$ is: $a \approx 5.6 \text{ cm}$

Would you like more details on this calculation, or do you have any questions? Here are some related questions that might interest you:

1. How would you find the length of side $b$ in this triangle?
2. What is the length of side $b$ if $\cos(47^\circ)$ is used?
3. Can you explain how the Pythagorean theorem applies to this problem?
4. What other trigonometric functions could be used to solve for side $a$?
5. How does the angle sum property of triangles verify the given angles?
6. What if angle $B$ was different? How would that affect the length of side $a$?
7. How do you find the area of this triangle?
8. How would you solve this problem using a different method?

Tip: Remember, in a right triangle, knowing one acute angle and the hypotenuse allows you to find all the other sides and angles using basic trigonometric functions.