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Real-World Applications of Trigonometry

Before beginning this tutorial, it may be helpful to review the tutorials on Right-Triangle Trigonometry, Graphs of Trigonometric Functions, and Graph Translation and Stretching.

Now we are ready to see how this trigonometry can be used in interesting ways.  Right triangle trigonometry can be used to solve many kinds of problems involving indirect measurement.  Let’s look at a few examples to see how:

Example 1: Viewing Angle to an Object

When two vertical objects of different heights m and n stand distance k apart and the tops of the objects aligned with the sun’s rays on a sunny day, the objects and the shadows they cast can be represented by a pair of similar triangles as shown below:

The shorter object (with height n) casts a shadow of length j; the larger object (with height m) casts a shadow of length k+j . Notice that . Using this model, we can use similar triangles to find the heights of objects.

Problem
A 5-foot tall woman stands 15 feet from a flagpole; she casts a shadow 7 feet long which ends at exactly the same point as the shadow of the flagpole. How tall is the flagpole?

Solution
From the picture above, if h represents the height of the flagpole, .  Multiplying both sides of this equation by 22 yields the result: feet.  That is, the flagpole is about 15.7 feet tall.

Angles of Elevation and Depression

We often do indirect measurement using angles of elevation and depression.  Angles of elevation are measured upward from the horizontal, while angles of depression are measured downward from the horizontal.

Problem
The angle of elevation from the bottom of a ski lift to the top of a mountain is 28°.  If a skier rides a distance of 900 ft. on this ski lift to get to the top of the mountain, what is the vertical distance d from the bottom of the ski lift to the top of the mountain?

Problem
A pilot must approach an airport at a descent angle (angle of depression) of 11° toward the runway.  If the plane is flying at an altitude of 3200 ft, at what distance d (in miles measured along the ground) should the pilot start the descent?

(remember that there are 5280 feet in one mile).

Other Indirect Measurement

Problem
A tent is supported by a cable stretched between two poles at a height of 80 inches.  The sides of the tent make an angle of 58° with the ground.  How wide is the tent at the bottom?

You can now see that trigonometry can help to find lengths that may not be directly measurable.