1. **State the problem:** We have two right triangles and need to find the primary trigonometric ratios (sine, cosine, tangent) for angle $A$ in each. 2. **Recall the primary trigonometric ratios for angle $A$ in a right triangle:** - $\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan A = \frac{\text{opposite}}{\text{adjacent}}$ 3. **Part (a): Triangle ABC with sides $AB=5$, $BC=12$, $AC=13$ and right angle at $B$** - Opposite side to $A$ is $BC=12$ - Adjacent side to $A$ is $AB=5$ - Hypotenuse is $AC=13$ Calculate: $$\sin A = \frac{12}{13}$$ $$\cos A = \frac{5}{13}$$ $$\tan A = \frac{12}{5}$$ 4. **Part (b): Right triangle with legs 8 and 15, right angle between them, angle $A$ at the vertex adjacent to side 8** - Opposite side to $A$ is 15 - Adjacent side to $A$ is 8 - Hypotenuse $= \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$ Calculate: $$\sin A = \frac{15}{17}$$ $$\cos A = \frac{8}{17}$$ $$\tan A = \frac{15}{8}$$ **Final answers:** - (a) $\sin A=\frac{12}{13}$, $\cos A=\frac{5}{13}$, $\tan A=\frac{12}{5}$ - (b) $\sin A=\frac{15}{17}$, $\cos A=\frac{8}{17}$, $\tan A=\frac{15}{8}$