Friday, September 11, 2020

Solving Application Problems (in Trigonometry)

I started this blog in 2009, was active for about 6 years, and then not so much for the past 5 years. I wrote two posts in the spring, both related to online teaching. We were all trying to learn how to teach well as we scrambled to do it while learning. I was happy to keep seeing my students online, and Zoom was our class. I used Canvas a little but not much.

Over the summer I learned a lot about effective online teaching. (I'm still not sure it can ever be nearly as effective as in-person, but...) I developed my Canvas shells for each course, and I started the semester readier than I had expected to be. My Canvas shells are not done. I created a "module" that orients students to online learning and my course. And I created a module for our first unit. The rest is still in progress.

Today I added a page for my trig students, on solving application problems. I want to share it here. (And I may share lots of my Canvas "pages" here, sometimes with modifications.)

Years ago, I modified George Polya's wonderful outline of problem solving steps. We start with that. It's a good idea to print it out, and turn to it whenever you're stuck. 

tree with shadow, pretty vs helpful

Draw a Diagram.

Always start by drawing a diagram. This step is vital, and is a major part of "Understanding the Problem".

Your diagram does not need to be artistically good. It does need to show relationships well. An artist might show my shadow going off at an angle. But for a math diagram, it is better to show the right angle involved, as a right angle.

In the diagrams on the right, the top drawing is prettier, and the shadow is more evocative, but the bottom drawing shows the right angle between a vertical object and its horizontal shadow, which is what will help you do your mathematical analysis.

Example (#22 in 2.4, page 93): If the angle of elevation of the sun is 63.4° when a building casts a shadow of 37.5 feet, what is the height of the building?

Draw your diagram now, labeling it with everything given and a variable for the value requested. (My drawing is below.)


















building with shadow, labeled



I labeled the height of the building h.











Write a Trig Equation.

In a simple problem, with only a few pieces of information this is all you need for the "Devising a Plan" step. We are given the value of the side adjacent (next to) the given angle, and we want to find the value of the side opposite the angle. (The hypotenuse is neither given nor asked for.) Which trig function uses adjacent and opposite? (Two of them do, but the one we use most of the time is...)














... LaTeX: \tan\theta=\frac{opp}{adj}, and this gives us  LaTeX: \tan63.4=\frac{h}{37.5}


Do a bit of algebra.

This is the "Carry out the Plan" step. To solve for h, we multiply both sides of the equation by 37.5:

LaTeX: 37.5\cdot\tan63.4=37.5\cdot\frac{h}{37.5}\:\:\Longrightarrow\:\:h=37.5\cdot\tan63.4=74.8857...

I pulled out my calculator for that last step (making sure it was in degree mode). Since our given length was given to tenths of a foot, I round, and give my final answer as 74.9 feet.


Check your Solution.

This is the "looking back" step on the handout. If we look at our diagram, does a height of about 75 feet seem reasonable? Well, the height seems bigger than the shadow, and maybe about twice as big, so yes, it seems reasonable.




If you get stuck on application problems, a good way to practice is to re-do problems that you've watched someone else do (perhaps on youtube). Try not to look at your notes. If you need to, go ahead and look. Do as much of the problem on your own as you can. If you looked at your notes at all, do it again the next day.

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