1. Problem 1: Find length AB in the triangle where opposite side $=8$ cm, angle $\theta = 60^\circ$, and adjacent side is AB.\n 2. Use the tangent function: $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$\n\n3. Substitute values: $$\tan 60^\circ = \frac{8}{AB}$$\n\n4. Calculate $\tan 60^\circ = \sqrt{3} \approx 1.732$. Thus, $$1.732 = \frac{8}{AB}$$\n\n5. Solve for AB: $$AB = \frac{8}{1.732} \approx 4.62\text{ cm}$$\n 6. Problem 2: Find angle $\theta = \angle BAC$ in a right triangle with hypotenuse $16$ cm, adjacent side $8$ cm.\n 7. Use the cosine function: $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{8}{16} = 0.5$$\n\n8. Find $\theta$ by inverse cosine: $$\theta = \cos^{-1}(0.5) = 60^\circ$$\n 9. Problem 3: Given triangle with sides adjacent $3$ cm, opposite $4$ cm, find angle $\theta$.\n 10. Use tangent: $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{4}{3} \approx 1.333$$\n\n11. Compute $\theta$: $$\theta = \tan^{-1}(1.333) \approx 53.13^\circ$$\n **Final answers:**\n1. $AB \approx 4.62$ cm\n2. $\theta = 60^\circ$\n3. $\theta \approx 53.13^\circ$