1. The problem asks to determine the value of $b$ in the function $h(\theta) = \cos(b(\theta + c))$ given the graph with peaks at $\left(\frac{\pi}{5}, 1\right)$ and $\left(\frac{7\pi}{10}, 1\right)$. 2. Recall that the cosine function has a period $T = \frac{2\pi}{b}$. The distance between two consecutive peaks (maximum points) is the period $T$. 3. Calculate the distance between the two given peaks: $$\frac{7\pi}{10} - \frac{\pi}{5} = \frac{7\pi}{10} - \frac{2\pi}{10} = \frac{5\pi}{10} = \frac{\pi}{2}.$$ 4. Since this distance is the period $T$, we have $$T = \frac{\pi}{2} = \frac{2\pi}{b}.$$ 5. Solve for $b$: $$\frac{\pi}{2} = \frac{2\pi}{b} \implies \cancel{\pi} \frac{1}{2} = \frac{2 \cancel{\pi}}{b} \implies \frac{1}{2} = \frac{2}{b} \implies b = 4.$$ 6. Therefore, the value of $b$ is $4$.