The value of $\displaystyle\int_0^{\pi/2} \sin x \, dx$ is:

Step-by-step Solution:

$$\int_0^{\pi/2} \sin x \, dx = \Big[-\cos x\Big]_0^{\pi/2}$$

$$= (-\cos(\pi/2)) - (-\cos(0))$$

$$= (-0) - (-1)$$

$$= 0 + 1 = \boxed{1}$$

Hence, Option C is correct.