In yesterday’s post, we saw that restricting the domain of to permits the definition of , shown in the purple graph.

Today, I want to give some pedagogical thoughts on teaching this concept to Preaclculus students.

1. Notice that, if the purple graph was completed upward or downward, anything more than a half-period would violate the vertical line test and thus fail to be a function. Thinking back to the original function, that’s another way of saying that the original sine wave violates the horizontal line test.

2. Restricting the domain to was a perfectly arbitrary decision. As shown above, there are plenty of other domains that would have worked acceptably. Only tradition requires us to choose . (By the way, finding an expression for the restriction of to, say, is a standard problem in a first course in real analysis.)

3. Since is typically used as shorthand for , some students will make the natural mistake of thinking that is just shorthand for , or . So I like to address this head-on when introducing inverse trigonometric functions for the first time to my Precalculus students.

4. Unlike other inverse functions, it can be a little tricky for students to draw the graph of by hand because the line of reflection actually is the linearization of at . In other words, is the first term of the Taylor series of at . (For more details, see https://meangreenmath.com/2013/07/24/taylor-series-without-calculus-2/ or https://meangreenmath.com/2013/07/06/reminding-students-about-taylor-series-part-6/) Therefore, as seen in the picture, the line is very, very close to the graph of for .

To assist students with accurately drawing by hand the graph of , I point out that the original function levels off horizontally at the points and . Therefore, after reflecting through the line , the graph of enters almost vertically through the points and .

5. Most Precalculus students are not savvy enough to appreciate the nuances of domain and range in the above definitions. Therefore, after illustrating the importance of choosing an interval that satisfies the horizontal line test, I’ll give the following ways of remembering what means:

“Arcsine of is an angle. It is the angle whose sine is equal to . And it’s the angle that lies between and .

OR

means that and .

6. Since and are inverse functions, it’s always true that and . However, and are not inverse functions, where is the full sine function . Therefore, it’s possible for to be something other than . For example,

This is analogous to our earlier observation involving the square root function, which was also defined by a restricted domain:

.

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