Lesson 1 of 12 · 8 min
Why the domain is cut
NCERT §2.1
A rider on the Ferris wheel texts a friend: 'I'm 32 m up.' Where on the wheel is the cabin? At that height there are two places, one on the way up and one on the way down.
The story this chapter follows: The fair on the school ground
The lesson in notes
In short
Chapter 1 showed that f has an inverse exactly when f is one-one and onto. sin x takes each value in [−1, 1] infinitely often: sin 0 = sin π = sin 2π = 0. So sine, as a map from R to [−1, 1], is onto but far from one-one, and has no inverse.
The repair is to keep only a piece of the domain on which the function rises (or falls) exactly once through its whole range. On that piece it is one-one and onto, and the inverse exists.
For sine, any of the intervals [−3π/2, −π/2], [−π/2, π/2], [π/2, 3π/2], … works: on each one sine runs through every value from −1 to 1 exactly once.
Each choice of interval gives a different inverse, called a branch. They share the domain [−1, 1] but return angles from different intervals.
One branch per function is agreed on and called the principal value branch. When no branch is named, the inverse trigonometric function means that one.
Once cut down, the inverse obeys the Chapter 1 rules: its domain is the range of the trimmed function and its range is the trimmed domain.
Inverse trigonometric functions turn up again in calculus, where several standard integrals come out as sin⁻¹ or tan⁻¹ of something.