Greenwood press

TRANSLATIONS
145
Transposition from C major to G major.
Left: A translation pattern from Pueblo pottery. Right: A glide reflection—
a translation alternated with a reflection.

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TRIANGLE TRIGONOMETRY
Trigonometry can be used to find unknown lengths or angle measurements.
In a situation involving right triangles, only a side length and an angle measure-
ment are needed to determine the length of an object. This information is useful
to engineers, because they can find large or hard-to-measure distances without
having to measure them. For example, the height of a flagpole or a tall building
can be determined using a measured distance from the pole and an angle of ele-
vation from the ground (see below).
Suppose you measure a distance 10 meters away from a flagpole along the
ground. You record an angle of elevation at that point equal to 40°, as depicted.
In right-triangle trigonometry, one of the following three ratios can be used to
find the flag height of the poll:
sin θ =
opposite
hypotenuse
cos θ =
adjacent
hypotenuse
tan θ =
opposite
adjacent
.
In this case, 
tan θ (pronounced tangent) should be used, because the opposite
side from the angle of elevation 
θ is unknown (the height of the flagpole), and
the adjacent side is the distance along the ground of 10 meters. Therefore the
height of the flagpole, approximately 8.4 meters, can be found by solving the
equation
tan 40
◦
=
flagpole
10
.
Sometimes the angle of elevation is recorded from an object above the ground,
such as a transit sitting on a tripod, as illustrated. If the angle measurement is not
taken from the ground, then the height of the tripod will need to be included in the
final calculation. In this case, if the transit is 1.5 meters off of the ground, then the
angle of elevation would be approximately 35° (see the following figure). 
The missing length will be approximately 6.9 meters, after setting up an
equation using the tangent function, 
tan 35
◦
=
length above transit
10
. To find the 
complete flagpole length, the height of the transit will need to be added to this
calculation in order to obtain the same answer calculated earlier.