1. **Problem statement:** In triangle DEF, right angled at E, with side DE = 50 cm and angle DEF = 17°, find the length of DF. 2. **Formula and rules:** In a right triangle, the side opposite an angle can be found using the sine function, and the hypotenuse can be found using cosine or sine. Here, angle DEF is at vertex E, so DE is adjacent to angle DEF, and DF is the hypotenuse. 3. **Using cosine:** Since DE is adjacent to angle DEF and DF is the hypotenuse, $$\cos(17^\circ) = \frac{DE}{DF}$$ 4. **Rearranging to find DF:** $$DF = \frac{DE}{\cos(17^\circ)}$$ 5. **Substitute values:** $$DF = \frac{50}{\cos(17^\circ)}$$ 6. **Calculate cosine:** $$\cos(17^\circ) \approx 0.9563$$ 7. **Calculate DF:** $$DF = \frac{50}{0.9563} \approx 52.29 \text{ cm}$$ **Final answer:** The length of DF is approximately 52.29 cm.