Hello, chickenwing!

Two lookout stations, which are 25 miles apart along the coast on a north-south line,

spot an approaching yacht.

One lookout station measures the direction to the yacht at N33°E,

and the other station measures the direction of the yacht at S62°E.

How far is the yacht from each lookout station?

How far is the yacht from the coast?

A sketch would certainly help . . . Code:

A *
| *
|62°* b
| *
| 85°* C
| *
25 | *
| *
|33°* a
| *
| *
|*
B *

The lookout stations are at A and B; the yacht is at C.

Since angle A = 62° and angle B = 33°, then angle C = 85°.

. . . . . . . . . . . . .a . . . . . . c . . . . . . . . . . 25·sin62°

Law of Sines: . ------- .= . ------- . → . a .= . ----------- . ≈ . 22.16 miles

. . . . . . . . . . . sin A] . . . . sin C . . . . . . . . . .sin85°

. . . . . . . . . . . . .b . . . . . . c . . . . . . . . . . 25·sin33°

Law of Sines: . ------- .= . ------- . → . b .= . ----------- . ≈ . 13.67 miles

. . . . . . . . . . . sin B . . . . sin C . . . . . . . . . .sin85°

The distance from the yacht to the coast is:

. . d .= .b·sin62° .= .13.67(sin 62°) .≈ .12.1 miles