Engaging students: Using right-triangle trigonometry

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cody Luttrell. His topic, from Precalculus: using right-triangle trigonometry.

A.1 Now that students are able to use right triangle trigonometry, there is many things that they can do. For example, they know how to take the height of buildings if needed. If they are standing 45 feet away from a building and they have to look up approximately 60 degrees to see the top of the building, they can approximate the height of the building by using what they know about right triangle trigonometry. Ideally, they would say that the tan(60 degree)= (Height of building)/(distance from building = 45). They can now solve for the height of the building. The students could also use right triangle trigonometry to solve for the elevation it takes to look at the top of a building if they know the distance they are from the building and the height of the building. It would be set up as the previous example, but the students would be using inverse cosine to solve for the elevation.

A.2 An engaging activity and/or project I could do would be to find the height of a pump launch rocket. Let’s say I can find a rocket that states that it can travel up to 50 feet into the air. I could pose this problem to my students and ask how we can test to see if that is true. Some students may guess and say by using a measuring tape, ladder, etc. to measure the height of the rocket. I would then introduce right triangle trigonometry to the students. After a couple of days of practice, we can come back to the question of the height of the rocket. I could ask how the students could find the height of the rocket by using what we have just learned. Ideally, I would want to here that we can use tangent to find the height of the rocket. By using altimeters, I would then have the students stand at different distances from the rocket and measure the altitude. They would then compute the height of the rocket.

D.1 In the late 6th century BC, the Greek mathematician Pythagoras gave us the Pythagorean Theorem. This states that in a right triangle, the distance of the two legs of a right triangle squared added together is equal to the distance of the hypotenuse squared ( $a^2+b^2=c^2$ ). This actually was a special case for the law of cosines ( $c^2=a^2+b^2-2ab\cos(\theta)$ ). By also just knowing 2 side lengths of a right triangle, one may use the Pythagorean Theorem to solve for the third side which will then in return be able to give you the six trigonometric values for a right triangle. The Pythagorean Theorem also contributes to one of the most know trigonometric identities, $\sin^2 x+\cos^2 x=1$ . This can be seen in the unit circle where the legs of the right triangle are $\sin x$ and $\cos x$ and the hypotenuse is 1 unit long. Because Pythagoras gave us the Pythagorean Theorem, we were then able to solve more complex problems by using right triangle trigonometry.