1. **Problem Statement:** Find the missing values in the first diagram: a) $x$ (cm), b) $b$ (cm), c) $\theta$ (degrees). 2. **Given:** One side is 43.2 cm, an angle is 38°, and sides $a$, $b$, and angle $\theta$ are unknown. 3. **Formulas and rules:** - Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ - Sum of angles in triangle: $$A + B + C = 180^\circ$$ 4. **Step 1: Find $\theta$** - Given angle 38° and right triangle implied, assume $\theta = 90^\circ - 38^\circ = 52^\circ$ (if right angle is present). 5. **Step 2: Find side $b$ using Law of Sines** - Let side opposite 38° be 43.2 cm. - Then $$\frac{b}{\sin 52^\circ} = \frac{43.2}{\sin 38^\circ}$$ - Calculate: $$b = \frac{43.2 \times \sin 52^\circ}{\sin 38^\circ}$$ - Using approximate values: $\sin 52^\circ \approx 0.7880$, $\sin 38^\circ \approx 0.6157$ - $$b = \frac{43.2 \times 0.7880}{0.6157} = \frac{34.0416}{0.6157} \approx 55.27 \text{ cm}$$ 6. **Step 3: Find side $x$ (assuming $x = a$)** - Using Law of Sines again: $$\frac{x}{\sin 38^\circ} = \frac{43.2}{\sin 38^\circ}$$ - Since $x$ is opposite 38°, and side 43.2 cm is opposite 38°, $x = 43.2$ cm (if same side), else more info needed. 7. **Final answers:** - $x = 43.2$ cm - $b = 55.27$ cm - $\theta = 52^\circ$ --- **Note:** The problem contains multiple diagrams and questions, but per instructions, only the first problem is solved here.