Trigonometric Maths Question

Two points A and B are 100m apart on the same side of a tower. The angle of elevation of A to the top of the tower is 20 degrees and the angle of elevation from B is 27 degrees. Find the height of the tower, to the nearest metre.

Please give me working out to your answer.

Draw a right triangle PQA, where PQ is tower and QA is base and
Mark a point B on QA so that AB = 100 m

Join BP.


let QB = x

cot(20) = (x + 100) /PQ

cot(27) = x / PQ

cot(20) - cot(27) = 1/PQ [ x + 100 - x ]

=> cot(20) - cot(27) = 100/PQ

PQ = 100 / [cot(20) - cot(27) ]

= 100 / (2.7475 - 1.9626)

PQ = 100 /0.7849

= 127 m

Draw a picture for the info given.

vertical leg is height of tower = h
shorter horiz leg (with 27º angle of elevation) = x
longer horiz leg (with 20º angle of elevation) = x + 100

tan 27 = h/x
h = x tan 27

tan 20 = h/(x + 100)
h = x tan 20 + 100 tan 20

x tan 27 = x tan 20 + 100 tan 20
x tan 27 - x tan 20 = 100 tan 20
x(tan 27 - tan 20) = 100 tan 20
x = 100 tan 20/(tan 27 - tan 20)
x = 250

h = 127