1. **State the problem:** We have two right triangles and need to find all six trigonometric ratios (sin, cos, tan, csc, sec, cot) for each angle $\theta$ given the sides. --- ### First triangle: - Adjacent side (base) = 28 m - Opposite side (height) = 21 m - Hypotenuse = $\sqrt{28^2 + 21^2} = \sqrt{784 + 441} = \sqrt{1225} = 35$ m **Trigonometric ratios:** 1. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{21}{35} = \frac{3}{5}$ 2. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{28}{35} = \frac{4}{5}$ 3. $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{21}{28} = \frac{3}{4}$ Reciprocals: 4. $\csc \theta = \frac{1}{\sin \theta} = \frac{35}{21} = \frac{5}{3}$ 5. $\sec \theta = \frac{1}{\cos \theta} = \frac{35}{28} = \frac{5}{4}$ 6. $\cot \theta = \frac{1}{\tan \theta} = \frac{28}{21} = \frac{4}{3}$ --- ### Second triangle: - Adjacent side = 7 cm - Hypotenuse = 13 cm - Opposite side = $\sqrt{13^2 - 7^2} = \sqrt{169 - 49} = \sqrt{120} = 2\sqrt{30}$ cm **Trigonometric ratios:** 1. $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{2\sqrt{30}}{13}$ 2. $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{13}$ 3. $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{2\sqrt{30}}{7}$ Reciprocals: 4. $\csc \theta = \frac{1}{\sin \theta} = \frac{13}{2\sqrt{30}} = \frac{13\sqrt{30}}{60}$ (rationalized) 5. $\sec \theta = \frac{1}{\cos \theta} = \frac{13}{7}$ 6. $\cot \theta = \frac{1}{\tan \theta} = \frac{7}{2\sqrt{30}} = \frac{7\sqrt{30}}{60}$ (rationalized) --- **Summary of second triangle results matching options:** - $\sin \theta = \frac{2\sqrt{30}}{13}$ (option 8) - $\cos \theta = \frac{7}{13}$ (option 2) - $\tan \theta = \frac{2\sqrt{30}}{7}$ (option 6) - $\csc \theta = \frac{13\sqrt{30}}{60}$ (option 10) - $\sec \theta = \frac{13}{7}$ (option 3) - $\cot \theta = \frac{7\sqrt{30}}{60}$ (option 4) All options correspond correctly to the calculated ratios. Final answers are given clearly with simplified fractions and rationalized denominators where applicable.