To solve this problem using the Law of Sines, let's go through the calculations step-by-step.

Step 1: Use the Law of Sines to Find ∠B

The Law of Sines states that: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Given:

• $\angle A = 45^\circ$
• $a = 22$
• $b = 25$

Substitute these values into the Law of Sines to find $\sin B$: $\frac{\sin 45^\circ}{22} = \frac{\sin B}{25}$

$\sin B = 25 \cdot \frac{\sin 45^\circ}{22}$

Since $\sin 45^\circ = \frac{\sqrt{2}}{2}$, we get: $\sin B = 25 \cdot \frac{\sqrt{2}/2}{22} = \frac{25\sqrt{2}}{44} \approx 0.801$

Taking the inverse sine, we find: $B \approx \sin^{-1}(0.801) \approx 53.1^\circ \text{ or } 126.9^\circ$

Since we are asked for an obtuse angle, $\angle B = 126.9^\circ$.

Step 2: Find ∠C

Using the fact that the sum of angles in a triangle is $180^\circ$: $C = 180^\circ - A - B = 180^\circ - 45^\circ - 126.9^\circ \approx 8.1^\circ$

Step 3: Find Side $c$

Using the Law of Sines again: $\frac{\sin A}{a} = \frac{\sin C}{c}$

$c = a \cdot \frac{\sin C}{\sin A} = 22 \cdot \frac{\sin 8.1^\circ}{\sin 45^\circ}$

$c \approx 22 \cdot \frac{0.141}{0.707} \approx 4.39$

Final Answers

• $\angle B \approx 126.9^\circ$
• $\angle C \approx 8.1^\circ$
• $c \approx 4.39$

Would you like further clarification on any of these steps?

1. How would the solution change if $\angle B$ was acute instead?
2. What if side $b$ were smaller than $a$?
3. How does the Law of Sines differ from the Law of Cosines?
4. How is the obtuse angle determined in ambiguous SSA triangles?
5. Why is the sum of angles in a triangle always $180^\circ$?

Tip: Always check for multiple angle solutions in SSA problems, as they may lead to ambiguous cases.