Greenwood press

146
TRIANGLE TRIGONOMETRY
The height of a tall object, such as a flagpole,
can be determined with trigonometry by
measuring a distance along the ground and
an angle of elevation.
Engineers use transits to measure
angles of tall or hard-to-reach objects.

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Right-triangle trigonometry can be used to determine an unknown angle
based on two lengths. For example, the navigator of a ship will try to minimize
the traveling distance by adjusting the direction of the boat to account for the
water’s current. If the current is moving parallel to the waterfront, then the speed
of the boat observed from land will be greater due to the push from the current.
Suppose that the ship is moving perpendicular to the shore at 40 feet per second
and is recording a land speed of 42 feet per second. 
The current will push the boat off course if it is trying to reach a destination
directly across the river. Using the cosine of the angle 
cos θ, the ship’s navigator
can determine the angle in which to rotate the boat so that it does not move off
course. The cosine function is used in this case, because the two measurements
known are the adjacent (the boat speed) and hypotenuse (the land speed) sides of
the right triangle. Substituting the given values in this relationship, the unknown 
angle of 17.8° can be found by solving the equation 
cos θ =
40
42
. To find an angle 
measurement, the inverse cosine of the ratio, or 
cos
−1
(40/42), needs to be
entered on the calculator. This means that if the boat moves straight towards its
journey, it will actually veer off course by 17.8°. If the boat is still headed straight
without accounting for the current, it will veer almost one-third of a mile off
course for every mile traveled. To avoid this problem, the ship’s navigator will
have to turn the boat 17.8° away from the perpendicular path and against the cur-
rent in order to travel directly across the river.
Applications of right-triangle trigonometry also exist in areas outside of sur-
veying and navigation. Air-traffic control at small airports must establish the
cloud height in the evening to determine if there is enough visibility for pilots to
safely land their planes. A light source directed at a constant angle of 70° towards
the clouds situated 1,000 feet from an observer, and the observer’s angle of ele-