The ability to solve right triangles has many applications in the real world.
Many of these applications have to do with two-dimensional motion, while others
concern stationary objects. We'll discuss both.

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Two-Dimensional Motion

Two-dimensional motion can be represented by a
vector. Every vector can be resolved into a
vertical and a horizontal component. When a vector is combined with its
vertical and horizontal component, a right triangle is formed.

Figure %: A vector and its component parts form a right triangle

Often the motion of a vehicle of some kind is modeled using a vector. With
limited information, using right triangle solving techniques, it is possible to
find out a lot about the motion of an object in a two-dimensional plane. For
example, if a boat goes 12 miles in a direction 31^{o} north of east, how
far east did it travel? If the boat began at the origin, the problem looks like
this in the coordinate plane:

Figure %: A boat's motion is modeled in the coordinate plane

*c* = 12 and

*A* = 31^{o}. Then

*b* = *c* cos(*A*) 10.29. So the boat went
slightly more than 10 miles east on its journey.

The motion of a projectile in the air can also be easily modeled using a right
triangle. The most common example of this is an airplane's motion. For
example, if an airplane takes off at an angle of elevation of 15^{o} and
flies in a straight line for 3 miles, how high does it get? 3 sin(15) .78. The plane climbs about .78 miles. These types of problems use the
terms angle of elevation and angle of depression, which refer to the angles
created by an object's line of motion and the ground. They can be
mathematically represented by a vector and a horizontal line, usually the *x*-
axis.

Figure %: Angles of elevation and depression

A zero degree angle of elevation or depression means that the object is moving
along the ground--it is not in the air at all. A 90 degree angle of elevation
is motion directly upward, whereas a 90 degree angle of depression is motion
directly downward.

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Stationary Objects

Stationary objects that form right triangles can also be examined and understood
by using right triangle solving techniques. One of the most common examples of
a right triangle seen in real life is a situation in which a shadow is cast by a
tall object. For example, if a 40 ft. tree casts a 20 ft. shadow, at what angle
from vertical is the sun shining?

Figure %: The shadow cast by a tree forms a right triangle

As the picture shows,

tan(*x*) = = . So

*x* = arctan() 26.6^{o}.

Whenever you use a right triangle to model a real-life situation, it is
immensely helpful to draw a picture or diagram of the situation. Then labeling
the parts of the right triangle is easy and the problem can be simply solved.