The six principal value branches, and the table that fixes them

What to assume they know

• The previous topic's result: each ratio has a family of admissible intervals and the chapter promotes one member of each family
• The standard angles and their ratios, from Class XI, including the surd values
• The identities that convert an angle to its supplement and to its negative
• Interval notation, including an interval with one interior point removed
• Radian measure, and reading a fraction of pi as an angle

What they should be able to do

• Reproduce the six-row table from memory, with the correct bracket shape in every cell
• Read a row of it as three facts: what the function accepts, what it returns, and which of the two anchor intervals the outputs sit in
• State the convention that fixes what an unqualified inverse symbol means
• Define the principal value of an inverse trigonometric expression
• Compute a principal value in three moves — name the answer, convert to a direct trigonometric equation, then select the unique solution inside the promoted interval
• Handle a negative argument correctly for all six functions, and say why three of them answer with a negative angle and three with an obtuse one
• Evaluate a sum of principal values by settling each term separately before adding
• Choose the correct interval or value in a multiple-choice item that tests only the table

Where it usually goes wrong

• "The inverse cosine of a negative number is negative." It is obtuse. Three of the six return into an interval that contains no negative angle at all, and the inverse cosine is one of them. This is the most common single error in the chapter and it is worth its own section.
• "The output interval for the inverse tangent is closed, like the inverse sine's." It is open at both ends, and the table prints round brackets to say so. Nothing at all maps to plus or minus half pi, because the tangent is undefined there.
• "Zero is removed from the inverse cosecant's outputs because the inverse cosecant of zero is undefined." Two different zeros. Zero is removed from the output interval because the cosecant of zero does not exist; the input side excludes everything strictly between minus one and one, and that is a separate fact in a separate column.
• "The inverse secant and the inverse cosecant accept the same numbers as the inverse sine." They accept the complement: everything at least one in size. Two rows of the table share a domain and the other four split into two more pairs; a student who has not read the input column will feed one half into the wrong function.
• "Principal value is just another name for the answer." It is the answer subject to a constraint. Every one of these expressions has infinitely many candidate answers, and the constraint picks one. Drop the constraint and the question has no single answer at all.
• "If the question does not say principal value, any branch will do." Item two of the Note says otherwise. An unqualified symbol means the promoted branch. That is the whole reason the table can be examined.
• "You can add the arguments first and take one inverse at the end." You cannot. Each term has to be settled inside its own row, with its own output interval, and only then added. Question 11 mixes three different rows in one expression precisely to punish the shortcut.
• "Two options in a multiple-choice item that differ only in bracket shape are the same option." They are not, and question 13 is built out of that difference. Read the brackets aloud when the item is shown.

Questions to check understanding

• Write out the six-row table with correct brackets, given only the function names
• Find the principal value of a stated inverse expression, showing the conversion to a direct trigonometric equation
• Find the principal value of an expression with a negative argument, and state which output interval forced the sign of the answer
• Evaluate a sum or difference of two or three principal values
• Choose the correct output interval for a named inverse function from four options differing in numbers or brackets
• Given a candidate angle and an inverse expression, decide whether the angle is the principal value and say why not if it is not
• Say which two of the six functions share a given domain

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 table (Part I p. 25). Read off the printed page, it has six rows in this printed order — inverse sine, inverse cosine, inverse cosecant, inverse secant, inverse tangent, inverse cotangent — with an arrow between the accepted set and the returned set. The accepted sets are: the closed interval from minus one to one, twice; the real line with the open interval from minus one to one removed, twice; the whole real line, twice. The returned sets are: the closed interval from minus half pi to half pi; the closed interval from zero to pi; the first of those with zero removed; the second with half pi removed; the first with both ends removed; the second with both ends removed. Every bracket shape in that list was read off the page image, including the round brackets on the last two rows.
• How the six rows are really three pairs (not in the book). Verified against the table: rows one, three and five all return into the interval anchored at minus half pi and half pi, trimmed three different ways; rows two, four and six all return into the interval anchored at zero and pi, trimmed the same three ways. The chapter's own row order puts the two members of each pair adjacent, so the structure is already on the page — it is simply never pointed out.
• The boxed Note (Part I p. 26), three numbered items. The first separates the inverse notation from the reciprocal notation. The second fixes the working convention: if no branch is named, the promoted one is meant. The third defines the principal value as the output lying in the promoted branch. Item two is the one that makes the table binding rather than merely informative, and it is the item students skip.
• Example 1 (Part I p. 26). The inverse sine of one over root two. Verified: the chapter's own method is to name the answer, turn it into the statement that the sine of that answer is one over root two, and then read off the unique angle inside the promoted interval. That angle is pi by four.
• Example 2 (Part I p. 26). The inverse cotangent of minus one over root three. Verified: the cotangent of the answer is minus one over root three; the cotangent of pi by three is one over root three, and subtracting an angle from a half turn flips the sign of the cotangent, so the answer is two pi by three, which does lie in the promoted open interval from zero to pi. Note that the chapter reaches for a Class XI identity here without naming it, and that the whole difficulty of the item is the minus sign.
• Exercise 2.1, questions 1 to 10 (Part I pp. 26–27). Ten principal values. Verified, all ten, from the table and the standard angles: the inverse sine of minus one half is minus pi by six; the inverse cosine of root three over two is pi by six; the inverse cosecant of two is pi by six; the inverse tangent of minus root three is minus pi by three; the inverse cosine of minus one half is two pi by three; the inverse tangent of minus one is minus pi by four; the inverse secant of two over root three is pi by six; the inverse cotangent of root three is pi by six; the inverse cosine of minus one over root two is three pi by four; the inverse cosecant of minus root two is minus pi by four.
• The negative-argument pattern (not in the book). Verified: of the four items above with a negative argument, the two whose functions return into the interval anchored at minus half pi and half pi — the inverse sine and the inverse tangent, and the inverse cosecant with them — answered with a negative angle, while the two whose functions return into the interval anchored at zero and pi — the inverse cosine, and the inverse cotangent of Example 2 — answered with an obtuse angle. That split is forced by the table and holds for every argument, not just these. The chapter states it nowhere.
• Exercise 2.1 questions 11 and 12 (Part I p. 27). Two sums. Verified: the first is the inverse tangent of one, plus the inverse cosine of minus one half, plus the inverse sine of minus one half; term by term that is pi by four, plus two pi by three, minus pi by six, which totals three pi by four. The second is the inverse cosine of one half plus twice the inverse sine of one half, that is pi by three plus twice pi by six, which totals two pi by three. Note that the instruction line above these two asks only for values, where the ten above asked for principal values; by item two of the Note these are the same request.
• Exercise 2.1 question 13 (Part I p. 27). Which set does the output of the inverse sine lie in? Four intervals are offered: from zero to pi closed; from minus half pi to half pi closed; from zero to pi open; from minus half pi to half pi open. Verified against row one of the table: the second. The two wrong pairs are exactly the two traps — the right numbers with the wrong bracket, and the right bracket with the wrong numbers.
• Exercise 2.1 question 14 (Part I p. 27). The inverse tangent of root three, less the inverse secant of minus two, with four options offered. Verified: the first term is pi by three. For the second, the secant of the answer is minus two, so its cosine is minus one half, and the unique angle in the promoted interval from zero to pi other than half pi is two pi by three. The difference is pi by three minus two pi by three, that is minus pi by three, which is the second option. This item is the negative-argument rule and the table, and nothing else.
• The Summary table (Part I p. 32). The same six rows, with three differences a student will notice: each function is written as an equation in y rather than as a bare symbol, the table gains a heading row naming the three columns, and the phrase promoting the output column is given in the heading rather than in a sentence above. The bracket shapes are identical to Part I p. 25 — checked cell by cell on the page image, including the round brackets on the last two rows.

Figures to have open

• The six-row table itself, built with the repo's DataTable component, in the chapter's printed row order. It is shown for most of this topic and must be a single persistent object that rows are highlighted in, not six separate cards. Content from Part I p. 25; layout the explanation's.
• One angle axis running from minus half pi to pi, reused in sections 4, 8 and 9, with the two anchor intervals shaded in two tones and the trims drawn as filled or hollow endpoints and interior holes.
• A card for section 11 showing the four offered intervals at a size where a square bracket and a round bracket are unmistakably different. This is the one place in the chapter where the typography carries the whole answer.
• No graph is needed in this topic. The chapter's figures on Part I pp. 20–25 belong to the next one.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 2 "Inverse Trigonometric Functions", §2.2 Basic Concepts, the collected table of domains and output intervals, Part I p. 25
• The boxed Note, items 1 to 3, Part I p. 26
• Examples 1 and 2, Part I p. 26
• Exercise 2.1, questions 1 to 14, Part I pp. 26–27
• Summary, the table and the principal value bullet, Part I p. 32