Maryland Math F.BF.B.4.e

MathGrades 9–12BUILDING FUNCTIONS

The Standard

Build inverse trigonometric functions.

Maryland College and Career Ready Standards for Mathematics (2025) · Build a function that models a relationship between two quantities.

Teacher's field guide

What Students Need to Do: Students choose intervals where sine, cosine, and tangent never repeat an output. They swap inputs and outputs on those intervals to build functions that return angles from ratios.
What Mastery Looks Like: Students use [−π/2, π/2] for sine, [0, π] for cosine, and (−π/2, π/2) for tangent. They sketch each inverse by swapping graph coordinates and explain why arcsin(1/2) = π/6, not 5π/6.
Common Misconceptions: Students may read arcsin(x) as 1/sin(x). They may give every angle with the same sine value instead of the angle in the chosen interval, or try arcsin(2).
How to Assess It: Exit ticket: Restrict y = sin x to [−π/2, π/2], sketch its inverse across y = x, and find arcsin(1/2). Explain why 5π/6 is not the output.

Lesson moves

On graph paper, plot five points of sine on [−π/2, π/2], fold along y = x, and mark the inverse points.
Ask students to write two sentences explaining why arcsin(1/2) is π/6 rather than 5π/6.
Make cards with inverse expressions and angle values; pairs match them, then check each match against the chosen angle interval.
Give students a 10-foot ladder reaching 8 feet up a wall; have them find its angle with the ground using arcsin(8/10).

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