Inverse Trig ISN and Evaluate

We set the stage with our "exploring inverse trig functions" and now we move into how we use them in this course. We take a few minutes at the start of class to summarize what we learned with the explore by filling in a little ISN booklet of three pages (one each for sine, cosine & tangent):
   

Recall that you can "undo" any trig operation with its inverse to find the angle (such as in solving right triangles or other triangle application problems). What I am asking students to do now is to find the range value for a given domain value of a trig inverse function. 

Very subtle concept here. Students have to be very solid with their trig ratios to be successful here. Basically what we are doing is finding the angle when we are given the ratio. This can be very difficult for some students.

What I do is go over a few problems with them such as

and refer to the unit circle visuals for the inverse trig function

   

I remind them that the domain of the inverse (for sine & cosine) is very limited. We can only evaluate values between [-1, 1]. When the ratio is negative I ask them questions to help them remember that the original sine ratio is the y-coordinate in the function and where is the y-coordinate negative in the restricted unit circle visual?

Basically students came up with these "thinking steps" for evaluating:
1) look at the ratio - what special or quadrantal angle results in this ratio?
2) now be careful - what is the sign of the ratio? If it's negative think about which inverse trig ratio we are using and where the negative ratios reside in the "unit circle connection" we drew above. Use that to determine the angle.

I also ask,

 "which inverse trig ratios can have negative angles in their domain? which can only have positive angles in their domain?"

I like to keep them thinking with this one

(I ask them "what is the domain of arcsine?".....and what is the numerical value for pi?)

After a few examples students practice "how to evaluate" inverse trig ratios with this foldable:

And then I work through a few "double evaluations" such as:
        

You've got to be careful with the second one as the inverse doesn't always just undo the operation with the result being the given angle. 

And then to keep them on their toes I give them one of these. All kinds of confusion zings around the room as students try to remember what special angle has that trig ratio. When I confirm that there isn't one, I tell them that we really don't need to know the angle. We are ultimately just finding the trig ratio for cosine of some mystery angle. 

with some prompting the hope is that someone will remember that the tangent ratio is y/x and the cosine ratio is x/r so all they need to do is figure out r with the Pythagorean relationship for x, y & r.

Finally students use this foldable on the page opposite to practice these evaluations.

this double evaluation goes on top of the evaluation page - I glue the evaluation page onto the notebook page and this page gets taped across the top creating a little flap over the evaluation page. 

I also have a worksheet of practice for students. Preview of first few problems:

All my files on Inverse Trig Functions can be found HERE (and this is everything - introduction sheets, evaluate sheets, ISN materials, answer keys)