To solve this triangle, we have:

• Side $a = 23.07$ cm
• Side $b = 10.75$ cm
• Angle $A = 30.2^\circ$

We need to determine whether there are possible solutions using the Law of Sines:

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

Step 1: Calculate $\sin B$

$\sin B = \frac{b \cdot \sin A}{a} = \frac{10.75 \cdot \sin 30.2^\circ}{23.07}$

Calculating $\sin 30.2^\circ \approx 0.502$:

$\sin B = \frac{10.75 \cdot 0.502}{23.07} \approx 0.234$

Now, find $B$ using the inverse sine function:

$B = \sin^{-1}(0.234) \approx 13.5^\circ$

Step 2: Check for possible solutions

Since $\sin B$ is positive and less than 1, there are two possible cases for angle $B$:

1. $B \approx 13.5^\circ$
2. $B' = 180^\circ - 13.5^\circ \approx 166.5^\circ$

Step 3: Calculate angle $C$

For each case, find angle $C$:

1. If $B = 13.5^\circ$, then: $C = 180^\circ - A - B = 180^\circ - 30.2^\circ - 13.5^\circ \approx 136.3^\circ$

2. If $B = 166.5^\circ$, then: $C = 180^\circ - 30.2^\circ - 166.5^\circ = -16.7^\circ$

The second case is not possible because angle $C$ cannot be negative.

Step 4: Calculate side $c$

Using the Law of Sines:

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

Substitute $a = 23.07$, $\sin A = 0.502$, and $\sin C \approx \sin 136.3^\circ \approx 0.732$:

$c = \frac{23.07 \cdot 0.732}{0.502} \approx 33.63 \text{ cm}$

Thus, there is only one possible solution for the triangle, making option B the correct choice.

Would you like further details or have any questions?

Related Questions:

1. How is the Law of Sines derived and why is it useful?
2. What are the conditions for a triangle to have two possible solutions?
3. How can you verify the correctness of your solution using trigonometry?
4. What is the ambiguous case of the Law of Sines?
5. How would the solution change if angle $A$ was obtuse?

Tip:

When solving triangles using the Law of Sines, always check for the ambiguous case, especially if given two sides and a non-included angle (SSA condition).