1. Let's first solve Example 2: Calculate the length of side AB. Given: Right triangle ABC with angle C = 50°, side BC = 9 cm and angle B is 90°. - Since angle B is 90°, side AB is the hypotenuse, side BC is opposite angle A, and side AC is adjacent to angle C. - Here, to find AB (hypotenuse), we use sine or cosine with respect to angle C = 50°. - Opposite side to angle C is AB, but since BC=9 is opposite angle A, we will use sine on angle C: $$\sin(50^\circ)=\frac{opposite}{hypotenuse}=\frac{BC}{AB}=\frac{9}{AB}$$ 2. Solve for $AB$: $$AB=\frac{9}{\sin(50^\circ)}$$ Calculate: $$\sin(50^\circ) \approx 0.7660$$ So, $$AB=\frac{9}{0.7660} \approx 11.75\text{ cm}$$ --- 3. Now solve Example 3: Calculate the length of the hypotenuse. Given: Right triangle with one angle 20°, the side adjacent is 12 cm, and the other is the right angle. - Side adjacent to 20° angle is 12 cm, hypotenuse unknown. - Use cosine definition: $$\cos(20^\circ)=\frac{adjacent}{hypotenuse}=\frac{12}{hypotenuse}$$ 4. Solve for hypotenuse: $$hypotenuse=\frac{12}{\cos(20^\circ)}$$ Calculate: $$\cos(20^\circ) \approx 0.9397$$ So, $$hypotenuse=\frac{12}{0.9397} \approx 12.77\text{ cm}$$ --- Final answers: - Length of side AB in Example 2 is approximately 11.75 cm. - Length of hypotenuse in Example 3 is approximately 12.77 cm.