1. **State the problem:** We need to find the two smallest positive values of $p$ such that $$\sin(4.08p + 12.40) \cdot \tan(2.45p + 17.17) = 0.$$ 2. **Understand the equation:** The product of two functions is zero if at least one of the functions is zero. So, either $$\sin(4.08p + 12.40) = 0$$ or $$\tan(2.45p + 17.17) = 0.$$ 3. **Solve for zeros of sine:** The sine function is zero at integer multiples of $\pi$: $$4.08p + 12.40 = k\pi, \quad k \in \mathbb{Z}.$$ Rearranging for $p$: $$p = \frac{k\pi - 12.40}{4.08}.$$ 4. **Solve for zeros of tangent:** The tangent function is zero at integer multiples of $\pi$: $$2.45p + 17.17 = m\pi, \quad m \in \mathbb{Z}.$$ Rearranging for $p$: $$p = \frac{m\pi - 17.17}{2.45}.$$ 5. **Find positive values:** We want the smallest positive $p$ values from both sets. - For sine zeros: find smallest $k$ such that $p > 0$: $$p = \frac{k\pi - 12.40}{4.08} > 0 \implies k\pi > 12.40.$$ Calculate $k$: $$k > \frac{12.40}{\pi} \approx 3.948.$$ So smallest integer $k = 4$. Calculate $p$ for $k=4$: $$p = \frac{4\pi - 12.40}{4.08} = \frac{12.5664 - 12.40}{4.08} = \frac{0.1664}{4.08} \approx 0.0408.$$ Next $k=5$: $$p = \frac{5\pi - 12.40}{4.08} = \frac{15.708 - 12.40}{4.08} = \frac{3.308}{4.08} \approx 0.8108.$$ - For tangent zeros: find smallest $m$ such that $p > 0$: $$p = \frac{m\pi - 17.17}{2.45} > 0 \implies m\pi > 17.17.$$ Calculate $m$: $$m > \frac{17.17}{\pi} \approx 5.466.$$ So smallest integer $m = 6$. Calculate $p$ for $m=6$: $$p = \frac{6\pi - 17.17}{2.45} = \frac{18.850 - 17.17}{2.45} = \frac{1.68}{2.45} \approx 0.6857.$$ Next $m=7$: $$p = \frac{7\pi - 17.17}{2.45} = \frac{21.991 - 17.17}{2.45} = \frac{4.821}{2.45} \approx 1.9678.$$ 6. **List all positive candidates:** From sine zeros: $0.0408$, $0.8108$ From tangent zeros: $0.6857$, $1.9678$ 7. **Find the two smallest positive values:** Sort: $0.0408$, $0.6857$, $0.8108$, $1.9678$ The two smallest positive values of $p$ are: $$\boxed{0.0408 \text{ and } 0.6857}.$$