Answer: A. $\dfrac{\sqrt{11}}{6}$

Explanation:1. Understand the properties of angles in a right triangle: Triangle RST is a right triangle. Let's assume $\angle T$ is the right angle ($90^\circ$). The sum of angles in a triangle is $180^\circ$. So, $m\angle R + m\angle S + m\angle T = 180^\circ$. If $\angle T = 90^\circ$, then $m\angle R + m\angle S + 90^\circ = 180^\circ$, which means $m\angle R + m\angle S = 90^\circ$. This is consistent with the given information: "the sum of the measures of angle R and angle S is $90^{\circ}$". This confirms that R and S are the two acute angles in the right triangle.

2. Recall the cofunction identity: For any acute angle $\theta$, $\sin(\theta) = \cos(90^\circ - \theta)$ and $\cos(\theta) = \sin(90^\circ - \theta)$. Since $m\angle R + m\angle S = 90^\circ$, we can write $m\angle S = 90^\circ - m\angle R$ and $m\angle R = 90^\circ - m\angle S$.

3. Apply the cofunction identity: We are interested in $\cos(S)$. Using the identity $\cos(\theta) = \sin(90^\circ - \theta)$, let $\theta = S$. $\cos(S) = \sin(90^\circ - S)$ Since $m\angle R = 90^\circ - m\angle S$, we have $90^\circ - S = R$. Therefore, $\cos(S) = \sin(R)$.

4. Use the given value of $\sin(R)$: We are given that $\sin(R) = \dfrac{\sqrt{11}}{6}$. Since $\cos(S) = \sin(R)$, we have: $\cos(S) = \dfrac{\sqrt{11}}{6}$.

5. Check the solution with the options: The value of $\cos(S)$ is $\dfrac{\sqrt{11}}{6}$. This matches option [A].

6. Conclusion: The value of $\cos(S)$ is $\dfrac{\sqrt{11}}{6}$.