The measures of two angles of a triangle are $35^{\circ }57^{\prime }{4}^{\prime \prime }$ and $39^{\circ }{2}^{\prime }56^{\prime \prime }$. The circular value (in radians) of the third angle is

First, let's find the sum of the two given angles:
$$\text{Sum}=(35^{\circ }57^{\prime }{4}^{\prime \prime })+(39^{\circ }{2}^{\prime }56^{\prime \prime })$$

Add the seconds: $4^{\prime \prime }+56^{\prime \prime }=60^{\prime \prime }=1^{\prime }$
Add the minutes: $57^{\prime }+2^{\prime }+1^{\prime }(\text{from seconds})=60^{\prime }=1^{\circ }$
Add the degrees: $35^{\circ }+39^{\circ }+1^{\circ }(\text{from minutes})=75^{\circ }$

Since the sum of all angles in a triangle is $180^{\circ }$, the third angle is:

$$\text{Third Angle}=180^{\circ }-75^{\circ }=105^{\circ }$$

To convert $105^{\circ }$ to radians (circular value):

$$\text{Radians}=105\times \frac{\pi }{180}=\frac{7\pi }{12}$$