1. **Problem:** Given a right triangle with sides 12 (adjacent to angle $\alpha$) and 5 (opposite to angle $\alpha$), find $\sin \alpha$, $\cos \alpha$, and $\tan \alpha$. 2. **Formula and rules:** - $\sin \alpha = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \alpha = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \alpha = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Find the hypotenuse:** $$ \text{hypotenuse} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 $$ 4. **Calculate $\sin \alpha$:** $$ \sin \alpha = \frac{5}{13} $$ 5. **Calculate $\cos \alpha$:** $$ \cos \alpha = \frac{12}{13} $$ 6. **Calculate $\tan \alpha$:** $$ \tan \alpha = \frac{5}{12} $$ **Final answer:** - $\sin \alpha = \frac{5}{13}$ - $\cos \alpha = \frac{12}{13}$ - $\tan \alpha = \frac{5}{12}$