Cutting a Function Down to Size Inverse Trigonometric Functions

An inverse function only exists for a function that is one-one (never sends two different inputs to the same output) and onto (its outputs cover its whole intended range). Sine fails the one-one test badly: sin(0), sin(pi), and sin(2pi) are all equal to 0, and in fact sin repeats every single one of its output values infinitely often, once for every added multiple of 2 pi. Ask 'which angle has a sine of 0.5' and there is no single correct answer, only an endless family of them: pi/6, 5pi/6, pi/6 + 2pi, and so on forever. The fix is to restrict sine's domain to one carefully chosen interval, small enough that within it, sine never repeats a value and hits every value in [-1,1] exactly once. Sine restricted to [-pi/2, pi/2] does exactly this, and it is this restricted version, not the original sine over all real numbers, that finally has a genuine inverse.

The inverse of sine restricted to [-pi/2, pi/2] is written sin inverse (also called arcsin), a function whose domain is [-1,1] (matching sine's range) and whose range is exactly [-pi/2, pi/2] (matching the restricted domain sine used). This particular choice of restricted interval, and the resulting range, is called the principal value branch; other, less commonly used intervals like [pi/2, 3pi/2] would also make sine invertible there, giving a different branch, but 'sin inverse' without further qualification always means the principal one. Cosine follows the identical logic but with a different restricted interval, [0, pi], since sine and cosine are not one-one over the same stretch of the number line; cos inverse therefore has domain [-1,1] and principal range [0, pi]. Graphically, since an inverse function's graph is always the mirror image of the original across the line y=x (interchanging every point's x and y coordinates), the graph of y = sin inverse x looks like the S-shaped curve of y = sin x, reflected along that diagonal line, restricted to just the principal branch.

The same restriction-then-invert logic applies to the other four trigonometric functions, each needing its own carefully chosen interval. Tangent, restricted to (-pi/2, pi/2), becomes one-one and onto over all real numbers, giving tan inverse a domain of all reals and a principal range of (-pi/2, pi/2). Cotangent, restricted to (0, pi), gives cot inverse the same domain (all reals) but a different range, (0, pi). Cosecant and secant are trickier, since their own ranges already exclude the interval (-1,1) entirely; cosec inverse ends up with domain R - (-1,1) and range [-pi/2, pi/2] excluding 0, while sec inverse has the same domain but range [0, pi] excluding pi/2 (the one point where secant itself is undefined). Every one of these six domain-range pairs is worth having as a single reference table, since a huge share of exam questions simply ask for the principal value of a specific input, and getting the range wrong (picking an angle outside the correct principal branch) is the single most common way to answer correctly in every other respect and still lose the mark.

The notation sin inverse x looks exactly like it could mean '1 divided by sin x', the way f(x) to the power -1 usually means 1/f(x) for an ordinary number. It does not. Here, the -1 is a label marking 'the inverse FUNCTION of sine', not an exponent, and (sin x) to the power -1, the genuine reciprocal, is a completely different object, more commonly written cosec x. This is purely a notational collision, the same symbol used for two different meanings depending on context, and the book itself flags it directly as a note worth remembering rather than assuming. When no particular branch is mentioned by name, sin inverse (and each of its five relatives) always refers to its principal value branch by default; the specific value that lands inside that principal branch is called, unsurprisingly, the principal value.

Expressions mixing an inverse trig function with an ordinary one, like sin inverse(2x times the square root of (1-x squared)), look intimidating but usually collapse with one well-chosen substitution. Let x = sinθ. Then 2x times root(1-x squared) becomes 2 sinθ times root(1-sin squared θ) = 2 sinθ cosθ = sin(2θ), a direct double-angle match. So sin inverse(2x root(1-x squared)) = sin inverse(sin 2θ) = 2θ = 2 sin inverse x, valid for x in [-1/root2, 1/root2] (the range where 2θ stays inside the principal branch). The general technique: spot the shape of a familiar trigonometric identity hiding inside the expression, substitute x = sinθ, or x = tanθ, or x = secθ depending on which identity the expression resembles, simplify using ordinary trig identities, then convert back. A second, more subtle example: cot inverse[1/root(x squared - 1)], for x>1. Let x = secθ. Then root(x squared-1) = root(sec squared θ - 1) = tanθ, so the whole expression becomes cot inverse(cotθ) = θ = sec inverse x, the simplest possible form the original messy fraction reduces to.

This chapter's own history note repeats almost word for word the same central fact the previous chapter in this thread already established: trigonometry as a systematic subject, and the sine function specifically, began in India, with Aryabhata, Brahmagupta, and both Bhaskaras doing the foundational work centuries before it reached Europe via the Arab world. What this chapter adds is a much more specific, narrower historical fact: the actual notation sin inverse x, cos inverse x, and the rest, was proposed far more recently, by the British astronomer Sir John F. W. Herschel in 1813, as a shorthand for the older, more cumbersome 'arc sine', 'arc cosine' phrasing. A separate, unrelated figure entirely, the Greek philosopher Thales, is credited with using shadow-ratio reasoning (not inverse trig functions as such) to measure the height of an Egyptian pyramid from its shadow alone, the same core idea Class 10's Applications chapter and this thread's own second chapter used for heights and distances. Three separate historical threads, ancient Indian trigonometry, a nineteenth-century notation choice, and an even older Greek measuring trick, all converge on the six functions defined in this one chapter.