Question 375078
Find the height of a flagpole that sits atop a hill, if it makes an angle
 of 122 degrees with the hillside, and the angle of elevation between the 
side of the hill to the top of the flagpole is 35 degrees at a distance of 120 feet.
:
We are going to use the law of Sines, 
:
Let the angle at the top of the pole to a point on the hillside = A
Then the side opposite this angle = a, which is given as 120'
:
Let the angle between the hillside point to the top of the pole = B (35 degrees)
Then the  side opposite this angle = b (the height of the pole
:
The angle between the pole and the hillside, C = 122 degrees
:
Find angle A: 180 - 122 - 35 = 23 degrees
:
{{{b/sin(B)}}} = {{{a/sin(A)}}}
Using our values
{{{b/sin(35)}}} = {{{120/sin(23)}}}
Cross multiply
sin(23)*b = sin(35)*120
Which is
.39073b = .573576*120
.39073b = 68.829
b = {{{68.829/.39073}}}
b = 176.155 ft is the height of pole above the top of the hill