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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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