Answer
$\theta = 111.4^{\circ}$
Work Step by Step
Let angle $A = 38^{\circ}$. Then the side $a$ opposite angle $A$ has a length of $a = 3.6+1.6 = 5.2$
Let angle $B$ be the angle at the center of the gear with radius 3.6. Then the side $b$ opposite angle $B$ has a length of $b = 2.7+1.6 = 4.3$
We can find angle $B$:
$\frac{a}{sin~A} = \frac{b}{sin~B}$
$sin~B = \frac{b~sin~A}{a}$
$B = arcsin(\frac{b~sin~A}{a})$
$B = arcsin(\frac{(4.3)~sin~(38^{\circ})}{5.2})$
$B = 30.6^{\circ}$
We can find angle $\theta$:
$A+B+\theta = 180^{\circ}$
$\theta = 180^{\circ}-A-B$
$\theta = 180^{\circ}-38^{\circ}-30.6^{\circ}$
$\theta = 111.4^{\circ}$
