Question

A ship is anchored off a long straight shoreline that runs north and south. From two observation points 17 miles apart on shore, the bearings of the ship are N 33° E and S 58° E. What is the distance from the ship to each of the observation points? (Round your answers to one decimal place.)

Answer 1

Refer to figure 1 (see attached) for the initial drawing. The observation points are at A and B. I placed point A south of point B. The ship is at point C. I added in a compass as well. 

As shown in figure 1, angle A is 33 degrees. You start by facing north and then you turn to face the eastward direction by 33 degrees. Angle B is 58 degrees. You start by facing south and then you turn 58 degrees to the east. The distance from A to B is 17 miles. Let's make c = 17. The lowercase c is opposite the upper case point C. Similarly, lowercase 'a' is opposite uppercase A; lowercase b is opposite uppercase B.

Once you have everything drawn out as shown in figure 1, we can use this info to find angle C
The three angles A,B,C add to 180 degrees
A+B+C = 180
33+58+C = 180
91+C = 180
91+C-91 = 180-91
C = 89
---------------------------------------------
Now use the law of sines to find the length of segment 'a'
a/sin(A) = c/sin(C)
a/sin(33) = 17/sin(89)
a = sin(33)*(17/sin(89))
a = 9.26027397983573
a = 9.3

Do the same for segment b
b/sin(B) = c/sin(C)
b/sin(58) = 17/sin(89)
b = sin(58)*(17/sin(89))
b = 14.419013720289
b = 14.4
---------------------------------------------

See figure 2 (attached) to see the fully solved triangle. When I refer to "solved triangle", I mean that we have found all of its sides and angles. In this case we have...

Angles:
A = 33 degrees
B = 58 degrees
C = 89 degrees

Sides:
a = 9.3 miles
b = 14.4 miles
c = 17 miles

Final answers are roughly 9.3 miles and 14.4 miles.