1. **Problem:** Find the missing side $x$ in a right triangle with angle $54^\circ$, adjacent side $22$, and hypotenuse $H$. 2. **Formula:** Use cosine ratio since adjacent and hypotenuse are involved: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Step:** Given adjacent $=22$ and angle $54^\circ$, hypotenuse $H$ is unknown, but $x$ is opposite side. To find $x$, use sine ratio: $$\sin(54^\circ) = \frac{x}{H}$$ 4. **Step:** From cosine, $$\cos(54^\circ) = \frac{22}{H} \implies H = \frac{22}{\cos(54^\circ)}$$ 5. **Step:** Substitute $H$ into sine equation: $$\sin(54^\circ) = \frac{x}{\frac{22}{\cos(54^\circ)}} = x \cdot \frac{\cos(54^\circ)}{22}$$ 6. **Step:** Solve for $x$: $$x = 22 \cdot \frac{\sin(54^\circ)}{\cos(54^\circ)} = 22 \cdot \tan(54^\circ)$$ 7. **Step:** Calculate numeric value: $$x \approx 22 \times 1.37638 = 30.28$$ **Answer:** $x \approx 30.28$ --- 2. **Problem:** Find missing side $x$ opposite angle $40^\circ$, adjacent side $15$, hypotenuse $H$ unknown. 3. **Formula:** Use tangent ratio: $$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$ 4. **Step:** Given adjacent $=15$, angle $40^\circ$: $$\tan(40^\circ) = \frac{x}{15} \implies x = 15 \cdot \tan(40^\circ)$$ 5. **Step:** Calculate numeric value: $$x \approx 15 \times 0.8391 = 12.59$$ **Answer:** $x \approx 12.59$ --- 3. **Problem:** Find missing side $x$ opposite angle $43^\circ$, adjacent side $31$, hypotenuse $A$ unknown. 4. **Formula:** Use tangent ratio: $$\tan(43^\circ) = \frac{x}{31}$$ 5. **Step:** Solve for $x$: $$x = 31 \cdot \tan(43^\circ)$$ 6. **Step:** Calculate numeric value: $$x \approx 31 \times 0.9325 = 28.91$$ **Answer:** $x \approx 28.91$