Trigonometric Ratios in Right Triangles
Lesson 2 of 6
Objective: SWBAT define trigonometric ratios and explain why they depend only on the acute angle measures of a right triangle.
Activate Prior Knowledge
In this section of the lesson, students work on Activating Prior Knowledge_Trig Ratios. I want students to warm up to the idea that corresponding pairs of side lengths in similar polygons are proportional. I also want to distinguish this fact from the corollary to this fact- that by properties of proportions, a pair of side lengths in one figure is proportional to the corresponding pair of side lengths in a similar figure. That is the purpose of #2 and #3 on the handout. It's an often-overlooked distinction, so I take some time to explain it for students.
To see if students understand my explanation I have them put it all together in the paragraph proof. They should be able to write something like:
The figures are similar because an image after dilation is similar to its preimage. Because corresponding pairs of side lengths in similar figures are proportional, BE/B'E' = ED/E'D'. Then by properties of proportions, BE/ED = B'E'/E'D'.
Finally, I want students to get used to solving for x in the structures x/a = c and a/x = c. These are the main structures we'll see when we use trigonometric ratios to solve for side lengths in right triangles. I point out that the cases are distinguished by the location of the variable of interest (numerator or denominator). In the case of a/x = c, I show the algebraic process of getting a/c=x but then I suggest a shortcut (MP8) by pointing out that when the variable of interest is in the denominator, we can simply transpose (i.e., switch) that variable with whatever is on the other side of the equation. This shortcut has longevity in the course so it is one that I continue to emphasize.
In this section, students will be working on Trig Ratios Data Collection and Inductive Reasoning. First they use a protractor to create a right triangle with a 40 degree angle. I announce to the class that we will be depending on the accuracy and precision of each other's work so take great care to be as accurate and precise as you can.
Next the students measure the side lengths of their triangles and calculate ratios of these lengths (sine, cosine, and tangent...even though we haven't defined them as such at this point). They will then compare data with a small group of their peers and make conjectures about what is generally true about the ratios of side lengths when we have triangle ABC with a 40 degree angle at A and a right angle at B.
Finally students will compile their ratios in a large table that I have created on the whiteboard. I will also be entering these data into an Excel spreadsheet or Google Sheet.
Once we have enough data on the board, I get the students attention and we begin a whole class analysis of the data. First I explain that in looking at a large data set such as this, the first thing we do is look for patterns and departures from patterns. What patterns do we notice? Is there anything that doesn't seem to fit with what we would expect based on the patterns? At this time, there are usually one or two students who realize they've made mistakes measuring, constructing, and/or calculating and need to correct them.
Next, students will typically express that the values of each ratio seems to be very close to one another regardless of the triangle that produced it. I ask, "Did everyone use the same triangle to calculate these ratios?...Were all of the triangles congruent?" "If not, then how do we explain how the ratios (BC/AC for example) were roughly the same for all of the different triangles we used? Since this is one of our essential questions, I stop for a pair-share at this point. Then I have several students report to the class what they discussed with their partner.
Next, I summarize/synthesize what hopefully came out during the student statements: So we know that by AA similarity all right triangles with 40 degree angles are similar. Human error aside, that means all of the triangles we produced are similar. Therefore their corresponding pairs of side lengths are proportional and by properties of proportions the ratio of any pair of sides in one triangle is the same as the corresponding ratio in all the other triangles. That's why BC/AC was about the same for all of the triangles.
Next, I let the students in on the fact that the ratios we calculated (there were many others we didn't calculate) were not arbitrarily chosen. They are special ratios, called trigonometric ratios, that are of interest to us when we deal with right triangles. I explain and have them label on their table,
BC/AC- sine 40 degrees (students label the heading of the second column "sin 40")
AB/AC- cosine 40 degrees
BC/AB- tangent 40 degrees
(Note: This is not the formal introduction to trig ratios; it's just a preliminary exposure. The formal introduction will come later in the lesson.)
Then I reiterate that theoretically, all of the triangles should have been similar and therefore we should have all gotten identical values for sin 40, cos 40 and tan 40 (I might pause here to have a quick pair share: Why didn't we all get the same values?). And then I ask the students if we had to make a conjecture as to what those true values of sin 40, cos 40, and tan 40 are, how would we go about doing that based on the data we have? I stop for a pair share to give all students the opportunity to think about this. Hopefully students will suggest taking the average, but if they do not, then I will explain why taking the averages makes sense. I then demonstrate how I can use Excel or Google Sheets to find the average values of the ratios.
Finally, I share with students that their predecessors have gone through the trouble of finding these trigonometric ratios for several possible right triangles and have recorded these in tables. Together we take a look at a trig table (either in the book or on handouts). I ask them to find sin, cos, and tan of 40 degrees so that they can be amazed that we actually calculated trigonometric ratios just by measuring side lengths and dividing them one by the other. At this time, I also have students locate the trig ratio buttons on their calculators (degree mode!) and see that this is also a way to retrieve the ratios we calculated.
Remembering the Ratios
In this section of the lesson, I want to formally introduce students to Trigonometry as the study of relationships in right triangles. I explain that there are three basic trig ratios we'll be dealing with in Geometry: sine, cosine, and tangent. I remind them that (as we saw in the previous section) these ratios depend only on the measure of an acute angle in a right triangle. So whenever we say sine, cosine, or tangent, it's always with respect to some acute angle of a right triangle.
We begin this section with a fun little PowerPoint presentation entitled (what else?) SohCahToa.
As I am having fun going through the presentation my objectives are:
- To stress the fundamental importance of considering point of view when discussing trigonometric ratios
- To define sine, cosine and tangent so that students understand the definitions
- To have students remember all three trigonometric ratios
- To have students make sense of the terms opposite and adjacent
So this is a lighthearted activity with lots of shameless repetition (SohCahToa) and emphasis on the importance of considering point of view (did I say that already?).
This section is a chance for students to practice, apply, and even extend what they have learned so far. I have my students work in pairs. That way they work on their own when they can and they have the reassurance of a peer to help when they need it.
Students work on Practice and Check for Understanding_Trig Ratios.
When students have had time to complete the first side of the handout, I call their attention to the front of the class so that I can demonstrate the answers to the front side. I do this to give studetns immediate feedback on their work and also in case there are any students who are having trouble accessing the material. Hopefully this modeling will allow them to access the second page of the handout.
As we near the end of the class period, provided students have had adequate time to work on the back side of the handout, I will go over the answers for the second side. As I have been walking around the classroom engaging with students, I have identified exemplars for #14. I select a few of these to read their responses the class while showing their paper under the document camera. Time permitting, I will have a student also show the method for #15. Otherwise, we'll save this for the next lesson on Solving Right Triangles Using Trigonometric Ratios.