1. **Problem statement:** Find the tangent of angle $B$ in a right triangle with vertices $A$, $B$, and $C$, where $AC=15$, $BC=17$, and angle $A$ is $90^\circ$. 2. **Recall the definition of tangent:** $$\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$$ For angle $B$, the opposite side is $AC$ and the adjacent side is $AB$. 3. **Find the missing side $AB$ using the Pythagorean theorem:** $$BC^2 = AB^2 + AC^2$$ $$17^2 = AB^2 + 15^2$$ $$289 = AB^2 + 225$$ $$AB^2 = 289 - 225 = 64$$ $$AB = \sqrt{64} = 8$$ 4. **Calculate $\tan(B)$:** $$\tan(B) = \frac{AC}{AB} = \frac{15}{8}$$ 5. **Final answer:** $$\boxed{\frac{15}{8}}$$ This is a proper fraction representing the tangent of angle $B$.