1. **Problem 1:** Given $A=40^\circ$, $B=80^\circ$, and $a=15$ cm, find $b$. 2. **Formula:** Law of Sines states $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}.$$ We use $$\frac{a}{\sin A} = \frac{b}{\sin B}$$ to find $b$. 3. **Step 1:** Calculate $b$ using $$b = \frac{a \sin B}{\sin A}.$$ Substitute values: $$b = \frac{15 \times \sin 80^\circ}{\sin 40^\circ}.$$ 4. **Step 2:** Evaluate sines: $$\sin 80^\circ \approx 0.9848, \quad \sin 40^\circ \approx 0.6428.$$ 5. **Step 3:** Substitute and simplify: $$b = \frac{15 \times 0.9848}{0.6428} = \frac{14.772}{0.6428}.$$ 6. **Step 4:** Calculate the division: $$b \approx 22.98 \text{ cm}.$$ --- 1. **Problem 2:** Given $A=55^\circ$, $B=45^\circ$, and $a=12$ m, find $c$. 2. **Step 1:** Find angle $C$ using triangle angle sum: $$C = 180^\circ - A - B = 180^\circ - 55^\circ - 45^\circ = 80^\circ.$$ 3. **Step 2:** Use Law of Sines: $$\frac{a}{\sin A} = \frac{c}{\sin C} \implies c = \frac{a \sin C}{\sin A}.$$ 4. **Step 3:** Substitute values: $$c = \frac{12 \times \sin 80^\circ}{\sin 55^\circ}.$$ 5. **Step 4:** Evaluate sines: $$\sin 80^\circ \approx 0.9848, \quad \sin 55^\circ \approx 0.8192.$$ 6. **Step 5:** Calculate: $$c = \frac{12 \times 0.9848}{0.8192} = \frac{11.8176}{0.8192} \approx 14.43 \text{ m}.$$ --- 1. **Problem 3:** Given $A=30^\circ$, $a=8$, and $b=10$, find $B$. 2. **Step 1:** Use Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} \implies \sin B = \frac{b \sin A}{a}.$$ 3. **Step 2:** Substitute values: $$\sin B = \frac{10 \times \sin 30^\circ}{8}.$$ 4. **Step 3:** Evaluate $\sin 30^\circ$: $$\sin 30^\circ = 0.5.$$ 5. **Step 4:** Calculate: $$\sin B = \frac{10 \times 0.5}{8} = \frac{5}{8} = 0.625.$$ 6. **Step 5:** Find angle $B$: $$B = \sin^{-1}(0.625) \approx 38.68^\circ.$$