As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points] (1) $\frac { 2 } { 3 }$ (2) $\frac { 8 } { 9 }$ (3) $\frac { 10 } { 9 }$ (4) $\frac { 4 } { 3 }$ (5) $\frac { 14 } { 9 }$
As shown in the figure, there is an isosceles triangle ABC with $\angle \mathrm { CAB } = \angle \mathrm { BCA } = \theta$ that is externally tangent to a circle of radius 1. On the extension of segment AB, a point D (not equal to A) is chosen such that $\angle \mathrm { DCB } = \theta$. Let the area of triangle BDC be $S ( \theta )$. What is the value of $\lim _ { \theta \rightarrow + 0 } \{ \theta \times S ( \theta ) \}$? (Here, $0 < \theta < \frac { \pi } { 4 }$) [4 points]\\
(1) $\frac { 2 } { 3 }$\\
(2) $\frac { 8 } { 9 }$\\
(3) $\frac { 10 } { 9 }$\\
(4) $\frac { 4 } { 3 }$\\
(5) $\frac { 14 } { 9 }$