3.7 Inverse Trigonometric Functions

Restricting domains to invert

Trig functions aren't one-to-one, so we restrict domains to create inverse trigonometric functions. arcsin(x) has domain [-1,1], range [-π/2, π/2]. arccos(x) has domain [-1,1], range [0, π]. arctan(x) has domain all reals, range (-π/2, π/2). These restricted ranges make each inverse a true function.

Evaluating inverse trig functions

arcsin(x) = θ means sin(θ) = x, where θ is the unique angle in [-π/2, π/2]. Similarly for arccos and arctan with their own ranges. When composing like sin(arccos(x)), draw a right triangle using the inverse's angle and given ratio, then find the missing side with the Pythagorean theorem to evaluate the outer function.

Modeling with inverses

Inverse trig functions let us solve for an angle when given a ratio of sides, useful in real-world contexts like angles of elevation or periodic motion. Since ranges are restricted, the inverse only returns one specific angle — always check that the answer is reasonable for the context (e.g., an angle must make physical sense).

Key terms

arcsin(x) domain & range

Domain: [-1, 1]; Range: [-π/2, π/2]

arccos(x) domain & range

Domain: [-1, 1]; Range: [0, π]

arctan(x) domain & range

Domain: all real numbers; Range: (-π/2, π/2)

Why restrict domains for inverse trig functions?

Trig functions are periodic and not one-to-one, so restricting the domain ensures each output angle is unique, making the inverse a true function.

Strategy for evaluating compositions (e.g., sin(arccos(x)))

Draw a right triangle representing the inverse function's angle using the given ratio, then use the Pythagorean theorem to find the missing side and evaluate the outer function.

Try an AP-style question

arcsin(1/2) = ?

1. 5π/6
2. π/3
3. 30
4. π/6