Answer:
about 113 meters
Step-by-step explanation:
If the straight-line distance is 1, then the distance around is ...
sin(30.6°) +cos(30.6°) ≈ 1.36978.
The 156 meters due east is the product of the straight-line distance and sin(30.6°), so we have ...
excess distance = (1.36978 -1)·(straight-line distance)
= 0.36978·(156 m)/sin(30.6°) = 113.3 m
The group walked about 113 meters farther to avoid the briar patch.
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More detail
The sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse. Here, that is ...
sin(30.6°) = (156 m)/(straight-line distance)
Multiplying by the denominator, and dividing by sin(30.6°), we get ...
(straight-line distance)·sin(30.6°) = 156 m
straight-line distance = (156 m)/sin(30.6°) . . . . . . the value used above