We are given the following conditions for a triangle:

• $a = 33$
• $c = 44$
• $\angle A = 39^\circ$

We need to solve for the two possible triangles that satisfy these conditions using the Law of Sines.

Step 1: Use the Law of Sines

The Law of Sines states that:

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

From the information provided, we can first calculate the value of $\frac{a}{\sin(A)}$:

$\frac{33}{\sin(39^\circ)} \approx \frac{33}{0.6293} \approx 52.5$

Now, we can use this to find angle $B$ (there are two possible cases for $B$).

Step 2: Solve for $\angle B$

Using the Law of Sines:

$\frac{33}{\sin(39^\circ)} = \frac{44}{\sin(B)}$

This simplifies to:

$\frac{33}{0.6293} = \frac{44}{\sin(B)} \quad \Rightarrow \quad \sin(B) = \frac{44 \times 0.6293}{33} \approx 0.8492$

Now, solving for $B$:

$B_1 = \sin^{-1}(0.8492) \approx 58.7^\circ$

Since the sum of angles in a triangle is 180°, we can calculate $C$:

$C_1 = 180^\circ - 39^\circ - 58.7^\circ = 82.3^\circ$

For the second possible angle $B_2$, we use the fact that $\sin(B) = \sin(180^\circ - B)$, which gives us the second solution for $B$:

$B_2 = 180^\circ - 58.7^\circ = 121.3^\circ$

Now, calculating $C_2$:

$C_2 = 180^\circ - 39^\circ - 121.3^\circ = 19.7^\circ$

Step 3: Solve for sides $b_1$ and $b_2$

Now, using the Law of Sines again to find the lengths of side $b$ for each case.

For $B_1 = 58.7^\circ$ and $C_1 = 82.3^\circ$:

$\frac{b_1}{\sin(58.7^\circ)} = \frac{33}{\sin(39^\circ)} \quad \Rightarrow \quad b_1 = \frac{33 \times \sin(58.7^\circ)}{\sin(39^\circ)} \approx \frac{33 \times 0.8572}{0.6293} \approx 45.4$

For $B_2 = 121.3^\circ$ and $C_2 = 19.7^\circ$:

$\frac{b_2}{\sin(121.3^\circ)} = \frac{33}{\sin(39^\circ)} \quad \Rightarrow \quad b_2 = \frac{33 \times \sin(121.3^\circ)}{\sin(39^\circ)} \approx \frac{33 \times 0.8387}{0.6293} \approx 45.0$

Final Answers

• $\angle B_1 \approx 58.7^\circ$
• $\angle B_2 \approx 121.3^\circ$
• $\angle C_1 \approx 82.3^\circ$
• $\angle C_2 \approx 19.7^\circ$
• $b_1 \approx 45.4$
• $b_2 \approx 45.0$

Tip:

When applying the Law of Sines, always check if the given conditions can create two possible solutions for the triangle, especially when an angle has two possible corresponding values.

Would you like more details on any of the steps?