Yesterday's lesson featured a lot of deep conceptual work about the need to restrict the domain of trig functions in order to produce an inverse that has exactly one output for each input. It was a lot of work, but we now we want to transition to thinking about how we actually find the value of expressions that involve inverse trig functions.
On question #7 on the worksheet, students will write out the equation in words. This will get students thinking about what they are actually finding. Just like when we work with logarithms, the symbols can get in the way and dilute the conceptual understanding of what exactly we are doing. In the previous unit, we continuously talked about how a logarithm is an exponent. In the same vein, you are going to have to remind students that the output of an inverse trig equation is an angle measure. And you are going to have to remind them a lot.
When you get something like sec-1(2sqrt(3)/3) = pi/6, students will get confused as to whether the 2sqrt(3)/3 or the pi/6 is the angle measure. Question 7 is an attempt to remedy this situation; I discuss what I am looking for from students in the video below.
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After working on the conceptual knowledge behind the functions, see if the students can evaluate some of them. Have them work on Question 8 with their tables. When discussing the answers, talk about how there are infinitely many angle measures that have a tangent of -1, but we can only choose the one that is in the interval [-90°, 90°] because that is how we restricted the function.
Here is an assignment to give some students practice with inverse trig functions and to extend their thinking about the concept. Something that is usually very challenging for students is to estimate csc-1(5/2) with their calculator. Here is where students have trouble with the notational conventions for inverse functions. The notation can obscure the meaning of what the students are actually asked to find. In talking about expressions like this, I stress that they are finding an angle whose cosecant is equal to 5/2. Since most calculators don’t have an inverse cosecant button, they will have to convert the expression to sin-1(2/5).
I find that many students will try to do 1/sin-1(5/2) since the sine and cosecant trig ratios are reciprocals. If enourage them to recall that sin-1(5/2) is an angle measure, and we want to take the reciprocal of the trig ratio, not the angle measure. This is a difficult concept for learners, but I find that continuously and clearly stressing that the answer to sin-1(5/2) is an angle measure helps students to internalize the idea.