To find the length of side $a$ in triangle $\Delta ABC$, we can use the Law of Sines. The Law of Sines states that:

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

We are given:

• $c = 720$ inches
• $\angle A = 74^\circ$
• $\angle B = 75^\circ$

To apply the Law of Sines, we first need to find the third angle $\angle C$. Since the sum of the angles in any triangle is $180^\circ$, we can find $\angle C$ as:

$C = 180^\circ - A - B = 180^\circ - 74^\circ - 75^\circ = 31^\circ$

Now, using the Law of Sines, we can solve for $a$:

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

Substituting the known values:

$\frac{a}{\sin 74^\circ} = \frac{720}{\sin 31^\circ}$

We can now calculate the values of the sines:

$\sin 74^\circ \approx 0.9613, \quad \sin 31^\circ \approx 0.5150$

Substitute these into the equation:

$\frac{a}{0.9613} = \frac{720}{0.5150}$

Now, solving for $a$:

$a = \frac{720 \times 0.9613}{0.5150} \approx \frac{692.736}{0.5150} \approx 1345.1$

Thus, the length of side $a$ is approximately 1345.1 inches (rounded to the nearest tenth).

Would you like more details on this process? Let me know if you have any questions!

Here are 5 related questions you might find interesting:

1. How can the Law of Cosines be used to solve for the sides or angles in a triangle?
2. What are some real-world applications of the Law of Sines in engineering or physics?
3. How can the angles of a triangle affect the length of its sides?
4. What happens to the sides of a triangle if the sum of the angles is not $180^\circ$?
5. How do you solve for the remaining angles when you know two sides and one angle in a triangle?

Tip: Always ensure your angle units are consistent (degrees or radians) when applying trigonometric formulas!