On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP. If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$. (Here, O is the origin, and $a , b$ are integers.) [3 points]
On the coordinate plane, let P $( t , \sin t ) ( 0 < t < \pi )$ be a point on the curve $y = \sin x$. Let circle $C$ be centered at P and tangent to the $x$-axis. Let Q be the point where circle $C$ is tangent to the $x$-axis, and let R be the point where circle $C$ meets segment OP.
If $\lim _ { t \rightarrow 0 + } \frac { \overline { \mathrm { OQ } } } { \overline { \mathrm { OR } } } = a + b \sqrt { 2 }$, find the value of $a + b$.\\
(Here, O is the origin, and $a , b$ are integers.) [3 points]