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Trigonometry is the study of
triangle measurement and calculations.
Trigonometry depends on special
functions called trigonometric
functions.
Motivation: Consider the following
right triangle (and included similar
triangles) that represent the shadow a
tree makes with the sun behind it.
Now the sun will cast a shadow at a
given time in the day a distance d from
the tree to the far vertex of the red
triangle. Using a surveying instrument
standing with your eyes and the
instrument at the circle you can
(without looking into the sun)
determine angle α1.
Since the triangles (black and red) are
similar we also know
α2. Since we can
measure the base of the triangle d and
using h for the height of the tree we
can solve for the height h using the
trigonometric ratio:
tan(
α2 ) = h / d
solving for h by multiplying by d
h = d * tan(
α2 )
so, if d = 100 ft and
α2 = 250 ,
tan(250) = 0.466,
then h = 100 ft * 0.466 =
46.6 ft
side note:
If you don't know you're trig table, or
have no calculator, a simpler method
using an intercept theorem from
geometry, if h1 is the height of the
instrument and d1 is length of the
shadow along the ground to the
instrument then
d1 / d = h1 / h, solving for h
we have h = (h1 * d) / d1
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