More on Finding Lengths of Segments
Lesson 2 of 3
Objective: Students will be able to continue to build on their understanding of finding lengths of segments using trig. Additionally they will investigate the relationship between the sine and cosine of complementary angles.
At the start of today's lesson the students will continue to work on the Lengths of Segments problem set that was introduced at the end of yesterday's lesson. The narrative for this problem set (from the previous lesson) includes:
Problem 7 asks that the students solve for both missing segments. I do not specify whether to use trigonometry or the Pythagorean Theorem, and I encourage discussion among the students, in order to ensure that the students understand that both methods work and provide similarly precise answers.
Problem 10 involves angle of depression. I discuss this with students as we encounter this, and compare it to angle of elevation. This is an opportunity to revisit parallel lines and alternate interior angles.
Problem 12 always generates interesting discussion about the height of the girl and its impact on the problem. This could be an interesting extension of the problem; since the water balloon sails beyond the girl – what would her height have to be for the balloon to hit her head?
I realize that some of these problems may seem slightly violent. The students usually find humor in these (as well as the goofy diagrams) and this helps to make the problems more engaging. Typically, my students are pretty far along as they enter class, but the opening segment gives them some time to finish and to talk about the problems and how they came up with an answer.
Investigation of Cofunctions
I present the Boat Problem on the SMART board and hand out a copy of it to my students. I ask the students to fill in their diagrams with the given information. Then, I ask students if they know anything else about this diagram. If necessary, I might ask, “We know the measures of two angles of the triangle – do we know the measure of the third angle?” Then I pose several questions:
Can you find the distance between A and B two different ways, using two different trig functions?
Students will hopefully use the sine of 48o and the cosine of 42o, and arrive at the same answer.
Why do you get the same answer?
If students need help in responding to this, I ask them to write their proportions using the names of the line segments. When they do so, using the substitution property, it is clear that sin 48 = cos 42. I ask the students to type sin 48 and cos 42 into their calculators, and they see that the values are the same. I ask them about these two angles. I say, “What word describes the relationship between 48o and 42o? Why is it that the two acute angles of every right triangle are always complementary?” I then ask them to use their calculators to investigate the sine and cosine of other complementary angle pairs.
More Word Problems
I present the students with Trig Word Problems. Today, the problems require the students to draw their own diagrams and to pay close attention to the structure of their diagram (MP7). I ask my students to work in groups and I walk around the room and watch for those students who might need help with developing the diagrams. For particularly challenged students, I provide scaffolding by giving them the same problem set with unlabeled diagrams provided. For those students who fly through the problem set and have class time remaining, I ask them each to develop his or her own trig word problem and then exchange it with another student.
Ticket Out the Door
I ask the students to complete and hand in the Ticket Out the Door. In this task the students are asked to find the lengths of as many segments as they can, rounded to the nearest tenth. Once a student finds the length of one line segment using trigonometry, there are a wide variety of approaches that can be employed to find the remaining segments. I am interested in seeing the different methods that students choose to employ, as well as the number of segments that each student is able to find correctly. I think this will give me an idea of where each student stands with regard to trig and solving right triangles.