Common Core Standards: Math
6. Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Many times, unruly teenagers do the opposite of what they're told to do. They're told to clean their room, do the dishes, and finish their math homework—so naturally, their room is a mess, the dishes are dirty, and their math homework remains incomplete. Not any of your students' math homework, of course.
The point is that these students love performing the inverse of what they're told. For their own safety and protection (and in part for the safety and protection of our retirement funds), we restrict their actions so that they can perform inverses all they want without repercussions. After all, there are worse things than a sink full of dirty dishes. Like incomplete math homework.
Well, looks like teenagers have more in common with their math homework than they thought. Inverse trig functions do the opposite of what trig functions do. They're like exponents and logarithms, sort of. If sin(x) = y, it only makes sense that sin-1(y) = x. For a restricted domain, anyway.
Students should be familiar with the concepts of "inverse" and "restrictions." Breaking curfew sums up both terms in one. Students might be less familiar with the notation of inverse trig functions.
Students should know that when we write sin-1(x), the -1 isn't a true exponent. It's just a convenient (and at times, inconvenient) way to indicate that we are using the inverse functions. Students can also denote inverse functions as arcsin(x) instead of sin-1(x). They're the exact same thing.
It's important for students to know that inverse trig functions are restricted to specific domains, sort of in the same way that logs and exponents are restricted. (Can we take the logarithm of a negative number? Spoiler alert: nope.)
Reviewing the definitions of domain and range will be helpful, and it never hurts to remind your students what makes a function a function. Combining the two may shed some light on inverse functions and why we have to restrict the domain to a region where it's always increasing or always decreasing.
They should remember that when swapping variables to form the inverse functions, the (restricted) domains of the trigonometric functions become the ranges of their corresponding inverse functions and vice versa. Kind of a "what's yours is mine and what's mine is yours" sort of deal.
Just because students are working with inverses doesn't mean they can throw ASTC out the window. It still applies, but to the input (x value) instead of the output. Essentially, it'll tell us what "quadrant" our y value is in.
Students will need to solve problems involving inverse trigonometric functions using their calculators. Even if their stubbornness tells them to do the opposite, they won't ignore that order.
On the other hand, students must understand how to correctly use their calculators to find inverse trigonometric functions. Since scientific calculators usually do not have inverse secant, cosecant, or cotangent buttons, they should convert these operations to their corresponding reciprocals before taking the inverse.
Students should come to realize that restrictions always have a purpose. In this case, a useful one. Once students practice with enough inverse trig functions, it should be nearly automatic. While their rooms will stay messy and the dishes won't be clean, at least their math homework will be done.