Why none of the six ratios has an inverse on its natural domain

What to assume they know

• The six trigonometric ratios as functions of a real variable, from Class XI
• The graph of each ratio over several turns, and the values it repeats
• Radian measure, and the standard angles as fractions of pi
• Interval notation, including a set written as one set minus another
• One-one and onto, and the fact from Chapter 1 that an inverse needs both
• Composition of two functions, and what the identity function does

What they should be able to do

• State the criterion the previous chapter fixed for a function to have an inverse, and apply it as a two-part test
• Write down each of the six ratios with the domain and the target set the chapter declares for it
• Explain why the domain of the tangent, cotangent, secant and cosecant functions is written as the real line with a set removed, and name the removed set in each case
• Distinguish the three shapes of output set that occur — a closed interval, the whole real line, and the real line with an open interval removed
• Argue from repetition, not from a picture, that every one of the six is many-to-one on its whole domain
• Say precisely which half of the two-part test fails, given that the chapter declares each target set to be the set of values actually taken
• Explain why a would-be inverse cannot pick an output, using one numerical value and its infinitely many pre-images
• Recover the standard facts about an inverse — the swap of domain and range, the two composition identities, and the double undo
• Separate the inverse from the reciprocal in the notation

Where it usually goes wrong

• "The trigonometric functions have no inverse, full stop." They have no inverse on the domain they arrive with. The entire rest of the chapter builds inverses by changing the domain, so an explanation that closes this topic on the flat claim has mis-set the next one.
• "They fail because they are not onto." As the chapter declares them on Part I p. 18, each target set is the set of values actually taken, so each is onto. The failure is that different inputs share an output. Say which half fails and the rest of the chapter reads as a repair; leave it vague and it reads as magic.
• "Sine repeats every two pi, so it takes each value twice per turn — I can just halve the domain." Halving is the right instinct and the wrong arithmetic. Inside one turn the value one half is taken at two different angles, so the interval you keep must be chosen so that it contains exactly one of them; which interval that is, is the subject of the next topic, and it is not simply half of a turn placed anywhere.
• "A raised minus one always means one over the thing." It does after a bracketed number, and it does not after a function name. The chapter's Note raises this before the first exercise because it is the single most expensive misreading in the chapter.
• "The domain of the tangent is the real line." It is the real line with the odd multiples of half pi taken out. Four of the six declarations on Part I p. 18 have a set subtracted, and a student who quotes the domain without the subtraction will later write an inverse tangent value at a point where the tangent does not exist.
• "Secant and cosecant take every real value, like tangent does." They skip the open interval from minus one to one. Both chapter lines print that hole explicitly, and it is the reason those two inverses have a domain shaped differently from all the others.
• "The inverse of a function and its reflection are two different ideas." They are the same idea in two representations, which the chapter says outright in the Remarks that follow. That is the fourth topic of this module; here just plant it.
• "Once we restrict, we have changed the sine function." A restriction has the same rule and a smaller domain. Nothing about the value of the sine at any surviving input changes. Students who believe the function was altered distrust every identity they already know.

Questions to check understanding

• State the condition under which a function has an inverse, and apply it to a named trigonometric function on its full domain
• Write the domain of a named ratio as the real line with a stated set removed, and justify the removal from the definition of that ratio
• Produce three distinct inputs sharing one output for a named ratio
• Say which of the two conditions fails for a named ratio, given the target set the chapter declares, and defend the answer
• Evaluate an inverse form and the corresponding reciprocal form at one angle and show that they differ
• Choose the interval in which the output of a named inverse function lies — the form of Exercise 2.1 Q13
• Given a function and its inverse, state the domain and range of each from the other

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The chapter's opening paragraph (§2.1, Part I p. 18). It does three things in order: it recalls that the undoing map needs the function to be both one-one and onto; it says that many functions are neither; and it says that the trigonometric functions, taken over their whole domains, are among them. Everything in this topic is an unpacking of that third clause.
• The six declarations (§2.2, Part I p. 18). Read off the printed page, the chapter sets them out as six lines, in this order and with these sets: sine, from the real line to the closed interval from minus one to one; cosine, the same; tangent, from the real line with the odd multiples of half pi removed, onto the whole real line; cotangent, from the real line with the whole multiples of pi removed, onto the whole real line; secant, from the real line with the odd multiples of half pi removed, onto the real line with the open interval from minus one to one removed; cosecant, from the real line with the whole multiples of pi removed, onto that same punctured line. Note: the chapter writes out the first name in full where every other line uses the short form. Say the names; do not spell the symbols.
• The removed sets, stated as conditions (§2.2, Part I p. 18). The chapter writes the excluded inputs as a condition on an integer: odd multiples of half pi for tangent and secant, whole multiples of pi for cotangent and cosecant. Verified: those are exactly the inputs at which the defining denominator vanishes — cosine is zero at the odd multiples of half pi, which kills tangent and secant; sine is zero at the whole multiples of pi, which kills cotangent and cosecant. The chapter states the two reciprocal relations later, on Part I pp. 21 and 22, and does not use them to explain the exclusions here.
• The value one half, and its pre-images (not in the book). Verified: the sine of pi by six is one half, and so is the sine of five pi by six, and so is the sine of every angle got from either by adding a whole number of full turns. That is an infinite list of inputs sharing one output, so no rule can look at the number one half and return the angle. The chapter asserts the failure and prints no such witness; supply this one, because it is the whole argument in a single line.
• What "not one-one and onto" is doing in that sentence (§2.1, Part I p. 18). It is a denial of the conjunction, not of both halves. Verified against the chapter's own declarations on the same page: each of the six lines names a target set that is exactly the set of values the ratio takes, so as printed, every one of the six is onto. The half that fails is injectivity, every time. This is worth thirty seconds because students who have been told "the trigonometric functions are not onto" will look for a value that is never taken and will not find one.
• The inverse machinery recalled (§2.2, Part I p. 19). For a one-one and onto map the chapter records: a unique undoing map exists; the domain of the undoing map is the range of the original and its range is the domain of the original; the undoing map is itself one-one and onto; undoing the undoing returns the original; and the two composites are the identity on the respective sets. All five are stated here without proof, having been settled in the previous chapter.
• The Note on notation (Part I p. 26). The chapter's boxed note opens by warning that a raised minus one written after the function name is not the same object as a raised minus one written after a bracketed value — the second is one divided by the sine. Verified: at pi by six the two disagree at once — the inverse sine of one half is pi by six, while one divided by the sine of pi by six is two. Use that pair; the chapter gives no numerical contrast.
• Exercise 2.1 Q13 (Part I p. 27). A multiple-choice item asking which set the output of the inverse sine lies in, offered as four intervals: the closed interval from zero to pi; the closed interval from minus half pi to half pi; the open interval from zero to pi; and the open interval from minus half pi to half pi. Verified against the chapter's own table on Part I p. 25: the answer is the second, the closed one from minus half pi to half pi. It belongs to this topic only as a forward pointer — the item exists because the choice made in the next three topics has to be made at all.
• The chapter frontispiece (Part I p. 18). The opening page carries a QR code marked with the Part I catalogue number and the chapter number, an epigraph attributed to Felix Klein about mathematics and self-evident things, and a portrait captioned Aryabhata with the dates 476 to 550 of the common era.
• The Historical Note (Part I p. 33). One page of narrative: trigonometry begins in India, passes to Arabia and then to Europe; four Indian mathematicians are named with dates; the raised-minus-one symbols themselves are credited to a nineteenth-century astronomer. Use two sentences of it as an end card. Note the printed spelling slip recorded below before showing the astronomer's name.

Figures to have open

• A full sine wave over at least three turns with a horizontal cut drawn at height one half, and every crossing marked. This is an added device; the chapter draws the wave on Part I p. 20 but draws no cut. It carries sections 1 and 6 and should be the same drawing both times.
• A six-row table of name, domain and target set for section 3. The content is the chapter's own Part I p. 18 list; the layout is added here. Build it with the repo's DataTable component.
• Two number lines for section 4, one with the odd multiples of half pi removed and one with the whole multiples of pi removed, drawn to the same scale so the two removed sets can be seen to interleave.
• A two-box arrow diagram for section 9, with the domain and range labels attached to the boxes rather than to the arrows, so the swap is visible when the second arrow is added.
• The Aryabhata portrait and the Klein epigraph on Part I p. 18 are the textbook's own. Use a caption card naming the portrait and its dates. Ignore the QR code.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 2 "Inverse Trigonometric Functions", §2.1 Introduction, Part I p. 18
• §2.2 Basic Concepts, the six declared functions, Part I p. 18, and the recalled inverse machinery, Part I p. 19
• The boxed Note, items 1 and 3, Part I p. 26
• Exercise 2.1, question 13, Part I p. 27
• Summary, the notation bullet, Part I p. 32; Historical Note, Part I p. 33