Solving Right Triangles

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.

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: 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. 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.

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? 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.