1. **State the problem:** Find the missing side $x$ in right triangles given an angle and one side. 2. **Recall formulas:** - Cosine relates adjacent side and hypotenuse: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$. - Tangent relates opposite and adjacent sides: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$. 3. **Solve each problem step-by-step:** **1)** Angle $68^\circ$, adjacent side $6$, find opposite side $x$ using tangent: $$\tan 68^\circ = \frac{x}{6}$$ Multiply both sides by 6: $$x = 6 \tan 68^\circ$$ Calculate: $$x \approx 6 \times 2.4751 = 14.9$$ **2)** Angle $71^\circ$, adjacent side $6$, find opposite side $x$ using tangent: $$\tan 71^\circ = \frac{x}{6}$$ Multiply both sides by 6: $$x = 6 \tan 71^\circ$$ Calculate: $$x \approx 6 \times 2.9042 = 17.4$$ **3)** Angle $42^\circ$, adjacent side $10$, find opposite side $x$ using tangent: $$\tan 42^\circ = \frac{x}{10}$$ Multiply both sides by 10: $$x = 10 \tan 42^\circ$$ Calculate: $$x \approx 10 \times 0.9004 = 9.0$$ **4)** Angle $63^\circ$, adjacent side $7$, find opposite side $x$ using tangent: $$\tan 63^\circ = \frac{x}{7}$$ Multiply both sides by 7: $$x = 7 \tan 63^\circ$$ Calculate: $$x \approx 7 \times 1.9626 = 13.7$$ **5)** Angle $74^\circ$, adjacent side $18$, find opposite side $x$ using tangent: $$\tan 74^\circ = \frac{x}{18}$$ Multiply both sides by 18: $$x = 18 \tan 74^\circ$$ Calculate: $$x \approx 18 \times 3.4874 = 62.8$$ **6)** Angle $69^\circ$, adjacent side $18$, find opposite side $x$ using tangent: $$\tan 69^\circ = \frac{x}{18}$$ Multiply both sides by 18: $$x = 18 \tan 69^\circ$$ Calculate: $$x \approx 18 \times 2.6051 = 46.9$$ **7)** Angle $70^\circ$, opposite side $14$, find adjacent side $x$ using cosine: $$\cos 70^\circ = \frac{x}{14}$$ Multiply both sides by 14: $$x = 14 \cos 70^\circ$$ Calculate: $$x \approx 14 \times 0.3420 = 4.8$$ **8)** Angle $19^\circ$, adjacent side $17$, find opposite side $x$ using tangent: $$\tan 19^\circ = \frac{x}{17}$$ Multiply both sides by 17: $$x = 17 \tan 19^\circ$$ Calculate: $$x \approx 17 \times 0.3443 = 5.9$$ **9)** Angle $37^\circ$, adjacent side $20$, find opposite side $x$ using tangent: $$\tan 37^\circ = \frac{x}{20}$$ Multiply both sides by 20: $$x = 20 \tan 37^\circ$$ Calculate: $$x \approx 20 \times 0.7536 = 15.1$$ **10)** Angle $38^\circ$, opposite side $x$, adjacent side $36$, find $x$ using tangent: $$\tan 38^\circ = \frac{x}{36}$$ Multiply both sides by 36: $$x = 36 \tan 38^\circ$$ Calculate: $$x \approx 36 \times 0.7813 = 28.1$$ **Final answers rounded to nearest tenth:** 1) $14.9$ 2) $17.4$ 3) $9.0$ 4) $13.7$ 5) $62.8$ 6) $46.9$ 7) $4.8$ 8) $5.9$ 9) $15.1$ 10) $28.1$