1. **Problem Statement:** Find the values of the trigonometric functions sin $\theta$, cos $\theta$, tan $\theta$, sec $\theta$, csc $\theta$, and cot $\theta$ for the acute angle $\theta$ in the given right triangles. 2. **Formulas and Definitions:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ - $\sec \theta = \frac{1}{\cos \theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$ - $\csc \theta = \frac{1}{\sin \theta} = \frac{\text{hypotenuse}}{\text{opposite}}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{\text{adjacent}}{\text{opposite}}$ 3. **Triangle 1 (Top-left graph):** - Opposite side = 3 - Adjacent side = 4 - Hypotenuse = 5 Calculate: $\sin \theta = \frac{3}{5} = 0.6$ $\cos \theta = \frac{4}{5} = 0.8$ $\tan \theta = \frac{3}{4} = 0.75$ $\sec \theta = \frac{5}{4} = 1.25$ $\csc \theta = \frac{5}{3} \approx 1.6667$ $\cot \theta = \frac{4}{3} \approx 1.3333$ 4. **Triangle 2 (Bottom-left graph):** - Opposite side = 3 cm - Adjacent side = 4 cm - Hypotenuse = 5 cm (by Pythagorean theorem) Values are the same as Triangle 1: $\sin \theta = \frac{3}{5} = 0.6$ $\cos \theta = \frac{4}{5} = 0.8$ $\tan \theta = \frac{3}{4} = 0.75$ $\sec \theta = \frac{5}{4} = 1.25$ $\csc \theta = \frac{5}{3} \approx 1.6667$ $\cot \theta = \frac{4}{3} \approx 1.3333$ 5. **Triangle 3 (Top-right graph):** - Opposite side = 8 m - Hypotenuse = 17 m - Adjacent side calculated by Pythagorean theorem: $$\text{adjacent} = \sqrt{17^2 - 8^2} = \sqrt{289 - 64} = \sqrt{225} = 15$$ Calculate: $\sin \theta = \frac{8}{17} \approx 0.4706$ $\cos \theta = \frac{15}{17} \approx 0.8824$ $\tan \theta = \frac{8}{15} \approx 0.5333$ $\sec \theta = \frac{17}{15} \approx 1.1333$ $\csc \theta = \frac{17}{8} = 2.125$ $\cot \theta = \frac{15}{8} = 1.875$ 6. **Triangles 4, 5, 6 (Center-right, Bottom-right, and last triangle):** - No side lengths given, so values cannot be calculated without additional information. **Final answers for the first triangle:** - $\sin \theta = 0.6$ - $\cos \theta = 0.8$ - $\tan \theta = 0.75$ - $\sec \theta = 1.25$ - $\csc \theta \approx 1.6667$ - $\cot \theta \approx 1.3333$