Inverse Trigonometric Functions

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SWBAT find the angles for any trig function given the measurements of the lengths or the ratio.

Big Idea

How can we find an angle measure without a protractor? Take a lesson from the Greeks and use inverse trigonometric functions!

Set the Stage

5 minutes

I begin class by reading some of the examples students gave in the previous lesson (No Angle Left Behind) or by giving examples of real-life applications of trig functions.  I make sure to finish with the Hydrogen Atom example from the Trig Example Problems resource.  I then ask students to consider individually what methods they would use to solve that problem.  (MP1, MP2, MP4)  I allow a few minutes for thinking then have students share their ideas with a partner.  I ask each student to explain his/her plan, then to critique each other’s plan (MP3).  I invite students to share with the class and allow brief discussion about which plan would be the easiest, best, etc.  This helps build student confidence for working with trig functions.  I then read the Refraction Example and ask students to follow the same procedure in planning a method to solve this problem. (MP1, MP2, MP4) I only allow a few minutes, gauging by the frustration/anxiety level of the students before I suggest that maybe we need a way to solve for missing angles.  That leads us to the main activity of the day.

Putting it into Action

40 minutes

Class Discussion (15-20 min): You will need copies of the Finding Angles with Trig handout for this section.  For most of this activity students will be collaborating in teams of 2-3.  Before beginning collaboration, I tell students that the trig ratios they’ve been using to find lengths of sides can also be used to find angles.  I give students the handout “Finding Angles with Trig” and, referring to the Refraction Example problem, I tell each team that they will have 20 minutes to create and implement a plan to solve the problem. (MP1, MP4)  I emphasize that there is always more than one way to find an answer and they can chose the method that looks best for them.  I also encourage them to read all of the information in the handout carefully before beginning their plan.  Many of my students jump immediately to the planning phase despite my warning to read the handout.  They usually get stumped pretty quickly and ask to be rescued, but instead of just telling them about inverse trig functions, I again advise them to read through the handout.  Technical reading is a major component of the new CCSS and also in most careers, so I’m trying to help my students develop this skill even if they don’t see its value at this time. (MP1, MP6)  At some point during their reading each team should realize that they need a scientific calculator, if they do not already have one.  I wait until they ask before distributing the calculators, another way of encouraging them to read the entire activity before diving in. During the work period I move around and make sure that no team is getting too far off course.  I also try to ask questions that will encourage deeper thinking about why they’re choosing a certain method or what they expect from a certain process.  For example I might ask "I see that you've set up some numbers to put into the calculator - Why/How did you choose those values?  What do you expect to get as a reasonable result?"

Student Presentations (12-15 min): At the end of 20 minutes, I ask each team to present their results, complete with explanation, sketches and calculations.  I generally draw names for which teams present in what order, but sometimes choose based on what I’ve heard and seen as I walked around the room.  Students not presenting are asked to evaluate the methods and solutions presented, with an emphasis on positive and productive comments. (MP3)

Wrap Up

5 minutes

To close this lesson I give each student a notecard and ask them to describe one method or process used by a classmate that they found particularly helpful in understanding the application of inverse trig functions and why it was helpful.  I remind students to use complete sentences and appropriate mathematical terms in their description. (MP6) This notecard is their ticket-out-the-door for the day and gives me insight into how they’re thinking about trig functions.