1. **State the problem:** We have a right triangle with sides $a=5$, $b=12$, and hypotenuse $c$. We need to find $c$, $\sin(A)$, $\cos(A)$, and $\tan(A)$ where $A$ is the angle opposite side $a$. 2. **Use the Pythagorean Theorem:** For a right triangle, $$c^2 = a^2 + b^2$$ 3. **Calculate $c$:** $$c^2 = 5^2 + 12^2 = 25 + 144 = 169$$ $$c = \sqrt{169} = 13$$ 4. **Recall definitions of trigonometric functions for angle $A$:** - $\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{a}{c}$ - $\cos(A) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{c}$ - $\tan(A) = \frac{\text{opposite}}{\text{adjacent}} = \frac{a}{b}$ 5. **Calculate $\sin(A)$:** $$\sin(A) = \frac{5}{13}$$ 6. **Calculate $\cos(A)$:** $$\cos(A) = \frac{12}{13}$$ 7. **Calculate $\tan(A)$:** $$\tan(A) = \frac{5}{12}$$ **Final answers:** - $c = 13$ - $\sin(A) = \frac{5}{13}$ - $\cos(A) = \frac{12}{13}$ - $\tan(A) = \frac{5}{12}$