Given curves $C _ { 1 }: y = \cos x$ and $C _ { 2 }: y = \sin \left( 2 x + \frac { \pi } { 6 } \right)$, which of the following conclusions is correct?
A. Stretch the horizontal coordinates of points on $C _ { 1 }$ to twice the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the right by $\frac { \pi } { 6 }$ units to obtain curve $C _ { 2 }$
B. Stretch the horizontal coordinates of points on $C _ { 1 }$ to twice the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the left by $\frac { \pi } { 12 }$ units to obtain curve $C _ { 2 }$
C. Compress the horizontal coordinates of points on $C _ { 1 }$ to half the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the right by $\frac { \pi } { 6 }$ units to obtain curve $C _ { 2 }$
D. Compress the horizontal coordinates of points on $C _ { 1 }$ to half the original length while keeping the vertical coordinates unchanged, then shift the resulting curve to the left by $\frac { \pi } { 12 }$ units to obtain curve $C _ { 2 }$