Inverse Trig Functions (Lesson 4.7)
Unit 4 - Day 13
Explain why and how the domains of sine, cosine, and tangent must be restricted to create an inverse function.
Use the restricted domains of the sine, cosine, and tangent, and reason to reason about the domains and ranges of the inverse functions.
Evaluate inverse trig expressions and equations
Quick Lesson Plan
With questions 1 and 2, we’d like students to understand that the inverse sine is not a function because the original sine function has repeated outputs, meaning it is not one-to-one. In order to make the inverse a function, they need to restrict the domain of the original, which becomes the range of the inverse. While the conventional range of the inverse function is [-(π/2), (π/2)], the students can also choose [(π/2), (3π/2)] if they’d like. We want them to be able to reason with the domains and ranges of the inverse, so giving them the freedom to choose their interval will help to make the connections.
For the debrief, make sure you explain how the restricted domains they chose in #4, #6, and #8 are the actual ranges of the inverse functions. If they chose a different interval than the conventional one, make sure you explain why we choose[-(π/2), (π/2)] instead of [(π/2), (3π/2)] for y = arcsin(x). I usually connect it back to the calculator and how it can only provide one answer, which is always between -90 and 90 degrees (for arcsin). Note to students that though this is the convention, it is a rather arbitrary choice, just as some notation is a choice made by the mathematical community that everyone agrees to adhere to.
They should also see the notation for inverse as arcsin, arccos, and arctan in addition to the usual “-1” superscript. Most inverse trig evaluating comes from the Unit Circle, so show the connection from the graphs of sine, cosine, and tangent to the quadrants of the Unit Circle in the Important Ideas.