Sine & Cosines - heading & distance

Question

You are at point A , to the west side of a tall mountain.  You are planning a hike around the mountain to point C, which is 9 miles east of A.  You start by hiking a distance of 6.6 miles on a heading of 65 degrees, which takes you to point B.  Find the heading and the distance from B to C.

Answer 1

Draw and label a diagram (forget about the mountain!)

m∠BAC = 25°

 

Use Law of Cosines to determine BC

BC² = 9² + 6.6² - (2)(9)(6.6)(cos 25″)

 

Use Law of Sines to determine m∠B

sin B / 9 = sin 25° / BC

Answer 2

draw a segment from A to C which is 9 miles

from A draw a segment to B at a bearing (angle) of 65º from due north

draw a segment from B to C

you have a triangle ABC where AC is horizontal, AB is slanted downward on the left and BC is slanted upward on the right

AC=9 miles, AB=6.6 miles and angle CAB=65º

use the law of cosines to find BC

BC²=9²+6.6²-2(6.6)(9)cos 65º

BC²=81+43.56-118.8cos65º

BCº=124.56-118.8*(0.422618)

BC²=124.56-50.2

BC²=74.36

BC=√74.36

BC=8.623 miles is the distance from B to C

knowing BC you can find the bearing from C to B by finding angle ACB

using the law of cosines to find angle ACB:

6.6²=9²+8.6²-2(9)(8.6)cosC

43.56=81+73.96-154.8cosC

doing the arithmetic you get:

-111.4/(-154.8)=cosC

cosC=0.7196

using arccos:

angle C=43.97º

90º-43.97º=46.03º

46.03º+90º=136.03º