## Unit 4 - Day 2

##### All Units
###### ​Learning Objectives​
• Understand that the angles in a right triangle are determined by the ratio of the sides

• Use the trigonometric ratios to find missing angles in a right triangle

###### Experience First

This lesson was a collaborative effort with my good friend and colleague, Lindsey Gallas, from Stats Medic. Ever since we have both used this approach for teaching inverse trig to our students, we have found that students have a much better understanding of the conceptual underpinnings of this topic.

The big idea of the activity is that whenever the sides of a right triangle are in a specific ratio, the angle is “set in stone”. Students look at three triangles where the opposite side is one-third of the hypotenuse. They reason that the missing angle must be 19.5˚ since all the triangles are similar and thus the angles must be congruent between triangles.

The driving question becomes: “Is there a way to “look up” the angle when given a particular ratio?” The answer is, of course, inverse trigonometry! I make a big deal of pulling up tables like this one and tell students they can just pull out their handy dandy chart and figure out what angle goes with their given ratio. They are actually quite satisfied with this answer but I go on to tell them that their calculator actually has this whole table stored for them and then I proceed to tell them how they can access those tables on their calculator. The benefit of this approach is that students don’t lose sight of the fact that the angles in a right triangle are determined by the ratio of sides because of similar triangles. We want to avoid students skipping too quickly to procedures and memorized algorithms (“when I want to find an angle I use the sine with the little negative 1 next to it”)

##### ​
###### Formalize Later

Although this lesson is very straightforward, use this as an opportunity to build good vocabulary and establish conceptual understanding of trigonometric ratios. These first two lessons of the unit are important stepping stones to understanding future ideas of the unit circle, special right triangles, and evaluating sine, cosine, and tangent for angles bigger than 90˚.

As much as possible, we want to link ideas back to what they’ve learned in Geometry so we don’t shy away from terms like similar triangles, congruent angles, and proportional sides.

A more formal lesson on inverse trig functions will come in lesson 4.7, so we avoid becoming overly technical about limited domains and ranges. Feel free to use informal language around what angle would have caused a particular ratio of sides instead of formal function notation.