Teach 180: Inverse Trig Functions (Day 168)

Note: After flooding the lawnmower multiple times, doing ironing, making dinner and spending time with my daughter, I went to bed and realized that I had not written in my blog for the day.  The events I am about to describe happened yesterday.

Most of my day was spent copying final exams for PreCalculus and organizing files in my classroom.  The one class I had today had only one student. (The other student in that class was at his sister's college graduation.)  We spent about 45 minutes of class time reviewing inverse trig functions.  We looked at the graph of f(x) = sin x and g(x) = sin^-1 x and realized that we would need to restrict f(x) so that it would be one-to-one.  Desmos made this very easy to see and very easy to check the validity of our chosen restriction for the domain of f(x) = sin x.  The graph of f(x) = sin x is in purple and the graph of its inverse is in black.  The dotted line is the line y = x to show that f(x) and its inverse are reflections of each other over the line y = x.  I added the points to the "ends" of the pieces to show that x and y coordinates are swapped when graphing a function and its inverse.

What about f(x) = cos x and g(x) = cos^-1 x?  Could we use the same restriction of -π/2 < x < π/2?  We used Desmos to see what would happen if we did this.

We could easily see that the restriction of -π/2 < x < π/2 was not correct. This was visible in two ways.  First, the restricted cosine function did not pass the horizontal line test.  Second, the red graph f(x) = cos x and the blue graph g(x) = cos^-1 x were not reflections of each other over the line y = x.  We decided to change our restriction to 0 < x < π and it worked!

I love how Desmos makes it easy to construct understanding and to play with the mathematics.  If we had chosen to do this discover by hand or with a graphing calculator, we would have been bogged down by the computations or button pushing.  Thank you Desmos for making discovery more accessible to everyone!