We are given a triangle $\Delta HIJ$ with the following information:

• Side $i = 7.3$ inches (opposite angle $I$),
• Side $h = 6.4$ inches (opposite angle $H$),
• Angle $\angle H = 57^\circ$.

To find the possible values of $\angle I$, we will apply the Law of Sines:

$\frac{i}{\sin I} = \frac{h}{\sin H}$

Step 1: Use the Law of Sines to solve for $\sin I$

$\frac{7.3}{\sin I} = \frac{6.4}{\sin 57^\circ}$ First, calculate $\sin 57^\circ$.

$\sin 57^\circ \approx 0.8387$

Now, substitute this value into the equation:

$\frac{7.3}{\sin I} = \frac{6.4}{0.8387}$

Simplify the right-hand side:

$\frac{7.3}{\sin I} \approx 7.63$

Now solve for $\sin I$:

$\sin I = \frac{7.3}{7.63} \approx 0.9567$

Step 2: Find $\angle I$

Using the inverse sine function:

$I = \sin^{-1}(0.9567) \approx 73.1^\circ$

Step 3: Check for another possible value of $\angle I$

Since the sine of an angle has two possible values (in the range $0^\circ$ to $180^\circ$), we check for the supplementary angle: $I' = 180^\circ - 73.1^\circ = 106.9^\circ$

Step 4: Validate the second angle

To see if $I' = 106.9^\circ$ is valid, recall that the sum of the angles in a triangle must be 180°. If $I' = 106.9^\circ$ and $\angle H = 57^\circ$, the remaining angle $J$ would be:

$\angle J = 180^\circ - 106.9^\circ - 57^\circ = 16.1^\circ$

This is a valid angle for a triangle, so both $73.1^\circ$ and $106.9^\circ$ are possible values for $\angle I$.

Conclusion:

The possible values of $\angle I$ are: $\boxed{73.1^\circ \text{ and } 106.9^\circ}$

Would you like more details on any of these steps?

Here are five related questions:

1. How do you apply the Law of Cosines in triangle problems?
2. How does the Law of Sines handle ambiguous cases (SSA triangles)?
3. What is the range of the inverse sine function, and why is it important?
4. How can you check if a triangle is obtuse, acute, or right-angled based on its angles?
5. How do you use trigonometric identities to solve triangle problems?

Tip: When using the Law of Sines, always check for the second possible angle due to the ambiguity (SSA case).