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ITF (Inverse Trigonometric Functions)

These are the inverses of the basic trigonometric functions like sin, cos, tan, etc.

1. Domain and Range of ITFs (Principal Values)

Inverse functions are defined only for specific domains (inputs) and ranges (outputs), called principal value branches, so that each inverse function gives a unique value.

FunctionDomainRange (Principal Value)sin⁻¹xcos⁻¹xtan⁻¹x (all real numbers) (open interval)cot⁻¹x (open interval)sec⁻¹x \ {π/2}cosec⁻¹x \ {0}

2. What is "Principal Value"?

A principal value of an inverse trigonometric function is a specific output (angle) chosen to make the function one-to-one.

Trigonometric functions are not naturally invertible on their full domain, because they are periodic.

We restrict their domains (usually the smallest symmetric interval around 0 where they are one-to-one) so we can define inverses.

This restricted range is called the principal branch or principal value range.

Example:

sin⁻¹(1/2) = π/6
Not 5π/6 or 13π/6 — even though sin(5π/6) = 1/2 too.
Why? Because π/6 is in the principal range .

Would you like a diagram showing these ranges visually on the unit circle?