1. **State the problem:** Calculate the length of the hypotenuse for each right triangle, given one angle (other than the right angle) and the length of a side adjacent to the right angle. 2. **Recall SOH CAH TOA:** Since we know the angle and the adjacent side, we use the cosine ratio: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Solve for hypotenuse:** $$\text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)}$$ --- **(a)** $\theta=35^\circ$, adjacent = 8 cm: $$\text{hypotenuse}_a = \frac{8}{\cos(35^\circ)}$$ Calculate: $$\cos(35^\circ) \approx 0.8192$$ $$\text{hypotenuse}_a = \frac{8}{0.8192} \approx 9.76 \text{ cm}$$ **(b)** $\theta=70^\circ$, adjacent = 3 cm: $$\text{hypotenuse}_b = \frac{3}{\cos(70^\circ)}$$ Calculate: $$\cos(70^\circ) \approx 0.3420$$ $$\text{hypotenuse}_b = \frac{3}{0.3420} \approx 8.77 \text{ cm}$$ **(c)** $\theta=41^\circ$, adjacent = 18 cm: $$\text{hypotenuse}_c = \frac{18}{\cos(41^\circ)}$$ Calculate: $$\cos(41^\circ) \approx 0.7547$$ $$\text{hypotenuse}_c = \frac{18}{0.7547} \approx 23.84 \text{ cm}$$ **(d)** $\theta=42^\circ$, adjacent = 33 cm: $$\text{hypotenuse}_d = \frac{33}{\cos(42^\circ)}$$ Calculate: $$\cos(42^\circ) \approx 0.7431$$ $$\text{hypotenuse}_d = \frac{33}{0.7431} \approx 44.41 \text{ cm}$$ **Final answers (3 significant figures):** - (a) 9.76 cm - (b) 8.77 cm - (c) 23.8 cm - (d) 44.4 cm