# The Tangent Function

Bobby is standing near a lighthouse. He measured the angle formed from where he stood to the top of the lighthouse and it was 30 degrees. Then he backed up 40 feet and measured the angle again and it was 25 degrees. Solve the height of the lighthouse.

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1. Label the original position as A and the second position as B, top of the lighhouse as P and its base as Q. ABQ is a straight line, AB = 40, angle B=25°, angle PAQ = 30°, AQ = x, and PQ = h

I used to teach this question is 2 ways:

1.
In triangle PBA, angle PAB = 150°
then angle BPA = 5°
by the sine law:
PA/sin25 = 40/sin5°
PA = 40sin25/sin5

in the right-angled triangle PAQ,
sin30° = h/PA
h = PAsin30 = (40sin25/sin5)(sin30) = 96.98..

method2:
in triangle PAQ, tan30 = h/x
h = xtan30
in triangle PBQ, tan25 = h/(x+40)
h = (x+40)tan25

thus: xtan30 = (x+40)tan25
xtan30 = xtan25 + 40tan25
x(tan30 - tan25) = 40tan25
x = 40tan25/(tan30-tan25)

h = xtan30
= (40tan25)(tan30)/(tan30-tan25)
= 96.98..

I consider method 1 as the easier of the two

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2. Or, method two simplifies a bit if you're not afraid of the cot(x) function:

h cot25 - h cot30 = 40

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