Understanding what inverse trig functions are, why they exist, and how they reverse the familiar trig ratios.
You know that sin(30°) = ½. But what if the question is reversed: "Which angle has a sine of ½?" This reverse question is exactly what inverse trigonometric functions answer.
If y = sin(x), then its inverse is x = sin⁻¹(y), read as "arcsin of y". It returns the angle whose sine equals the given value.
The six inverse trig functions correspond one-to-one with the six standard trig functions: arcsin, arccos, arctan, arccosec, arcsec, arccot.
sin⁻¹(x) is NOT the same as 1/sin(x). The ⁻¹ here denotes the inverse function, not a reciprocal power. The reciprocal of sin is written as cosec(x).
Alternative notation: Instead of sin⁻¹(x), you may also see arcsin(x) — both mean exactly the same thing.
For a function to have an inverse, it must be one-to-one (injective) — each output must come from exactly one input. However, trig functions are periodic and repeat values infinitely.
For example: sin(30°) = sin(150°) = sin(390°) = ½. If we asked for sin⁻¹(½), there would be infinitely many answers! To fix this, we restrict the domain to a specific interval where the function is one-to-one, called the principal value branch.
Inverse trig functions are only well-defined because we agree to restrict the original function to a specific interval. This selected interval is the principal value branch — covered in detail on Page 3.
These direct values are essential. Memorise them — they appear constantly in problems.