Understand that inverse trigonometric functions input ratios and output angles. The input and output values are switched from their corresponding trigonometric functions.
Explain why and how the domains of sine, cosine, and tangent must be restricted to create an inverse function.
Evaluate inverse trig expressions.
Quick Lesson Plan
In question 1, students are prompted to notice that there are many angles on the unit circle or in the domain of the sine function that output a value of -1/2. This becomes a problem when we want to evaluate the inverse sine function such as in question 3. We want students to understand that the relation that maps each sine ratio to its angle 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 function, which now provides a restricted range for the inverse function. While the conventional range of the inverse function is , the students can choose any one-to-one interval, like [(π/2), (3π/2)] if they’d like for question 5. The goal is to build conceptual understanding around why the domain needs to be restricted, and how the restricted interval is chosen. The margin notes will formalize that the conventional restriction is [-(π/2), (π/2)].
Can we create a function that inputs -½ and outputs 7π/6, -π/6, 11π/6, and 19π/6? Why or why not?
What’s the difference between solving a trig equation and evaluating an inverse trig function?
Why did you choose this interval to restrict the sine (cosine, tangent) function?
What happens when you evaluate sin^-1(-0.5) on your calculator?
For the debrief, make sure you explain that the restricted domains they chose in #5, #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).You can have students evaluate the inverse sine function on their calculator for a negative value like -0.3 to validate that only one answer is given, and that the angle is given as a negative value, rather than between 3π/2 and 2π. 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.
In the QuickNotes, provide the alternate inverse trig notation (arcsin(x), arccos(x), and arctan(x)) in addition to the usual “-1” superscript.
The Check Your Understanding questions focus on understanding the restricted domain and range of the inverse trig functions. In question 2, students may be quick to say “true” to both statements since inverse functions undo each other, without realizing that a value like 5π/6 is not in the range of arcsin(x).