Greenwood press

148
TRIANGLE TRIGONOMETRY
Determining the cloud height.

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tion from the base of the hill to its peak, and then repeats the measurement at a
given distance away, the law of sines can be used to find the height of the hill.
Actually, it can first be used to find the length along the side of the hill, and then
right-triangle trigonometry can be used to find the hill’s height. In this case, a
surveyor takes measurements 
c = 1, 000 feet apart and measures angles of ele-
vation to the tip of the hill equal to 
m < B = 75° and m < A = 43°. The fol-
lowing equation to find the length alongside the hill, 
a, can be set up using the 
law of sines: 
sin 32
◦
1,000
=
sin 43
◦
a
.
The 32° angle opposite the 1,000 foot distance can be found by using the fact
that the sum of the angles in a triangle is equal to 180°. This length of 
a, approx-
imately 1,287 feet, can help engineers determine the amount of railway needed
to build a funicular to transport materials, or the amount of cable needed to build
a gondola line for skiing. Since a right triangle is in the diagram, right-triangle
trigonometry can be used to find the hill’s height. Solve the equation 
sin 75° =
h
1,287
to determine the height of the hill, 
h, which is approximately 1,243 feet.
That is a length equal to about four football fields, but straight up in the air!
The law of cosines is a theorem used in triangle trigonometry to find the
measurement of a side when two sides and an included angle are given, or to find
the measurement of an angle when three sides are given. For example, a public-
works contractor can determine the amount of cement needed to pave a new road
that intersects two other intersecting roads in town (to form a triangle), as shown
below. 
In this case, the contractor needs to determine the angle formed between the
existing roads, 
m

C, and the location of the intersection of the other two roads
in order to predict the distance of the new road. Since the distance traveled is pro-
portional to the amount of cement used, the formula 
c
2
= a
2
+ b
2
− 2ab cos C
will help determine the amount of cement needed to connect the roads, where 
a,
b, and c are sides of the triangle (the length of the roads), and C is the angle in-
cluded between the existing roads 
a and b. A similar type of investigation would
also be needed for bridge designers or tunnel developers.