We are given the following triangle problem:

• $C = 61^\circ 20'$
• $c = 31.8$
• $b = 23.6$

We are tasked with solving the triangle by determining the remaining angle $B$, angle $A$, and side $a$. To do this, we can use the Law of Sines:

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

Step 1: Solve for $\sin B$

Using the Law of Sines, we can solve for $\sin B$:

$\frac{23.6}{\sin B} = \frac{31.8}{\sin 61^\circ 20'}$

First, calculate $\sin 61^\circ 20'$:

$\sin 61^\circ 20' \approx \sin 61.333^\circ \approx 0.8763$

Now, substitute this value into the equation:

$\frac{23.6}{\sin B} = \frac{31.8}{0.8763}$

Simplify:

$\frac{23.6}{\sin B} \approx 36.29$

Now, solve for $\sin B$:

$\sin B \approx \frac{23.6}{36.29} \approx 0.6502$

Step 2: Find $B$

Now take the inverse sine of $0.6502$ to find $B$:

$B \approx \sin^{-1}(0.6502) \approx 40^\circ 30'$

Step 3: Find angle $A$

Since the sum of angles in a triangle is $180^\circ$, we can find $A$ as follows:

$A = 180^\circ - C - B$ $A = 180^\circ - 61^\circ 20' - 40^\circ 30' \approx 78^\circ 10'$

Step 4: Solve for side $a$

Using the Law of Sines again, we can solve for side $a$:

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

Substitute the known values:

$\frac{a}{\sin 78^\circ 10'} = \frac{31.8}{0.8763}$

First, calculate $\sin 78^\circ 10'$:

$\sin 78^\circ 10' \approx 0.9787$

Now, substitute and solve for $a$:

$\frac{a}{0.9787} = 36.29$

$a \approx 36.29 \times 0.9787 \approx 35.52$

Conclusion:

• $B \approx 40^\circ 30'$
• $A \approx 78^\circ 10'$
• $a \approx 35.52$

Thus, the correct choice would be A. Would you like further details or have any questions?

Related questions:

1. How does the Law of Sines apply to obtuse triangles?
2. What happens if $\sin B$ exceeds 1 during calculations?
3. How can the Law of Cosines be used to solve this triangle instead?
4. How can rounding errors affect the final result in trigonometric calculations?
5. What are the possible solutions for an ambiguous case in the Law of Sines?

Tip:

Always check if the triangle has multiple solutions by considering the ambiguous case, especially when using the Law of Sines.