Greenwood press

TRIANGLE TRIGONOMETRY
147
Boats need to turn an angle by 
θ against the current
in order to account for the force of it so that they can
head in the most direct path towards the shore.
A transit can be used to measure the angle of elevation to help
determine the height of a tall object, such as a flagpole. The
sum of the height of the transit and the leg of the right triangle
along the flagpole represents the total height of the flagpole.

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vation
θ to the spotlight in the clouds, are sufficient information to determine the
cloud height (see below). 
In this situation, the equations 
tan θ =
h
y
,
tan 70° =
h
x
, and 
x + y = 1, 000
can be used to find the cloud height, 
h. Planes can safely land if the cloud height
is above 1,000 feet, with horizontal ground visibility of at least three miles.
The pilot can also use right-triangle trigonometry to determine the moment
when a plane needs to descend towards the airport. If the plane descends at a large
angle, the passengers may feel uneasy due to a quick drop in altitude and also
may not adjust well to changes in pressure. Consequently, the pilot tries to antic-
ipate the opportunity to descend towards the airport at a small angle, probably
less than 5°. Based on the plane’s altitude, air-traffic control at the airport can
determine the point at which the plane should begin to descend. With a descent
angle of 3° and altitude 
a, the plane should start its approach at a distance of
tan 3
◦
a
feet away from the airport, assuming that the plane descends at the same 
angle until it reaches the ground.
Construction workers can determine the length of a wheelchair ramp based
on restrictions for its angle of elevation. For example, suppose an office needs to
install a ramp that is inclined at most 5° from the ground. If the incline is too
great, it would be difficult for handicapped people to move up the ramp on their
own. Based on this information, the architect and construction workers can deter-
mine the number of turns needed in the ramp so that it will fit on the property
and stay within the angle-of-elevation regulations. In addition to wheelchair
ramps, a similar equation can be set up to determine the angle by which to pave
a driveway so that an automobile does not scrape its bumper on the curb upon
entering and leaving.
All triangle applications finding unknown sides or angles, however, are not
always situated in settings where a right triangle is used. In these cases, either the
law of sines or law of cosines can be applied. One example of applying the law
of sines is to find the height of a hill or a mountain, since it is unlikely that one
will be able to find the distance from the base of a hill or mountain to its center,
as shown in the following figure.
The law of sines states that the ratio of the sine of an angle to the side length
of its opposite side is proportional for all opposite angle and side pairs. That is,
in triangle ABC, 
sin A
a
=
sin B
b
=
sin C
c
. If a person measures an angle of eleva-