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Lesson 147: Solving Word Problems with Trigonometry
In this lesson we'll learn how to solve word problems using the trigonometry that we learned in the last lesson.
Before starting this lesson, be sure to review the previous lesson on trigonometry, and make sure that you fully understand it. This lesson will not make sense otherwise.
Here is a typical word problem that can be solved using trigonometry. A fireman rests his ladder against a building, making a 57° angle with the ground. The bottom of the ladder is 28 feet from the base of the building. How long is the ladder?
To
solve these problems, always make a diagram and label all of the
information that you know. Also label the information that you
need to know. See the picture at left for how I did this.
Notice how this word problem really translated into a standard right
triangle trig problem.
The first thing to determine is what trig ratio we should use to solve this problem. We have an angle, we have an adjacent side to that angle, and we need to know the hypotenuse. Referring to SOH-CAH-TOA, we can see that the cosine will help us. We know that cos 57 = 28/x. Using a calculator, we can determine that cos 57 = 0.545 (rounded). Using basic algebra, it is easy to determine that x = 51.4 (rounded). That means that the ladder is 51.4 ft long. Always check the question to see if it specifies that your answer should be rounded in a particular way.
Note that if we were given other information in the problem, we might have had to use the sine or the tangent to get our answer, and we might have had to do slightly different things with our algebra to solve for the unknown value.
Let's try one more problem which introduces an easy new concept. An pilot of an airplane in flight looks down at a point on the ground that is some distance away. The angle of depression is 28°, and the plane's altitude is 1200 meters. What is the distance from the pilot to the point on the ground.
First, we define angle of depression as the angle at which the pilot is looking down from an imaginary horizontal line. If he was looking straight ahead, the angle would be 0°. If he was looking straight down, the angle would be 90°. If he was looking halfway between straight ahead and straight down, the angle would be 45°, and so on.
Let's
make a diagram. We have to be careful, and remember that the
altitude is drawn straight down, perpendicular to the ground.
To draw the angle of depression, we must first draw an imaginary
line extending straight ahead from the plane, and measure the angle
downward from that line.
Once we do that, we can label the angle of elevation as shown. That is just the angle that is from the point on the ground to the plane. The angles of depression and elevation are always equal, since they are really alternate interior angles. Our unknown, x, is really the hypotenuse of the triangle. That is the distance from the pilot to the point on the ground.
Now, we have an angle (the one inside the triangle), we have an opposite side, and we need to know the hypotenuse. The ratio to use is sine. Sin 28 = 1200/x. Looking up the value of sin 28, and using basic algebra, we can see that x = 2556 meters (rounded).
These problems are tricky, and require lots of practice. Do as many as you can from your textbook, and contact me if you run into difficulties.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.