1. Given that $\sin \theta = \frac{5}{13}$ and $\theta$ is acute, find: (a) $\tan \theta$ (b) $\csc \theta$ (c) $\sin \theta$ (d) $\cos \theta$ (e) $\cot \theta$ --- **Step 1: Understand the problem** We are given $\sin \theta = \frac{5}{13}$ and $\theta$ is acute (meaning $0 < \theta < 90^\circ$). **Step 2: Recall the Pythagorean identity and definitions** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}}$ - $\csc \theta = \frac{1}{\sin \theta}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ **Step 3: Find $\cos \theta$ using Pythagoras** Since $\sin \theta = \frac{5}{13}$, opposite side = 5, hypotenuse = 13. Using Pythagoras theorem: $$\text{adjacent} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$$ So, $$\cos \theta = \frac{12}{13}$$ **Step 4: Calculate each required value** (a) $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{5/13}{12/13} = \frac{5}{12}$ (b) $\csc \theta = \frac{1}{\sin \theta} = \frac{1}{5/13} = \frac{13}{5}$ (c) $\sin \theta = \frac{5}{13}$ (given) (d) $\cos \theta = \frac{12}{13}$ (calculated) (e) $\cot \theta = \frac{1}{\tan \theta} = \frac{12}{5}$ --- **Final answers:** (a) $\tan \theta = \frac{5}{12}$ (b) $\csc \theta = \frac{13}{5}$ (c) $\sin \theta = \frac{5}{13}$ (d) $\cos \theta = \frac{12}{13}$ (e) $\cot \theta = \frac{12}{5}$