1. **Problem:** Find the length of side $x$ in a right triangle where the angle is $54^\circ$ and the adjacent side is 22. 2. **Formula:** Use the sine definition: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$. 3. **Step:** Given $\sin(54^\circ) = \frac{x}{H}$ and adjacent side $22$, we use the Pythagorean relation or tangent to find $x$. 4. **Using tangent:** $\tan(54^\circ) = \frac{x}{22}$, so $x = 22 \times \tan(54^\circ)$. 5. **Calculate:** $x = 22 \times \tan(54^\circ) \approx 22 \times 1.37638 = 30.28$. 6. **Answer:** The length of side $x$ is approximately $30.28$. --- 2. **Problem:** Find $x$ in a right triangle with angle $40^\circ$, adjacent side 15, and hypotenuse 12.6. 3. **Formula:** Use tangent and cosine relations. 4. **Step:** $\tan(40^\circ) = \frac{x}{15}$ so $x = 15 \times \tan(40^\circ)$. 5. **Calculate:** $x = 15 \times 0.8391 = 12.59$. 6. **Answer:** $x \approx 12.59$. --- 3. **Problem:** Find the opposite side $x$ in a right triangle with angle $43^\circ$ and adjacent side 31. 4. **Formula:** $\tan(43^\circ) = \frac{x}{31}$. 5. **Step:** $x = 31 \times \tan(43^\circ)$. 6. **Calculate:** $x = 31 \times 0.9325 = 28.91$. 7. **Answer:** $x \approx 28.91$.