Shifts by a quarter, a half and a whole turn all fall out of the same two results

What to assume they know

• One distance calculation on the unit circle yields the cosine of a difference — results 3 and 4 of §3.4, and the substitution trick that turns one into the other
• Coordinates on the unit circle extend the ratios to every real number — what the sine and the cosine come to at each quadrantal angle, since those are the only numbers this topic ever substitutes
• Which of the six stays positive where, and what happens at the quarter turns — negating the input, listed as results 1 and 2
• Substituting one expression for a variable throughout an identity

What they should be able to do

• Derive result 5 by substituting a quarter turn into result 4
• Derive result 6 from result 5 by applying it a second time
• Explain why the sine and cosine of complementary inputs exchange values
• Derive result 7 for the sine of a sum, naming the results used at each step
• Derive result 8 by the same negation used for result 4
• Produce any entry of result 9 by substitution, without consulting a table
• Explain why quarter-turn shifts swap the two functions and half-turn shifts do not
• Derive the half-turn behaviour of the tangent, and connect it to the repetition claimed on p. 55

Where it usually goes wrong

• "There are eight formulas here, or sixteen with the other functions." There are none. There is one expansion and a short list of quadrantal values to substitute into it. A student who can do the substitution can also handle the shifts Exercise 3.3 demands and result 9 never prints — the three-quarter turn in question 9, and the eighth and three-eighth turns in questions 6, 7 and 11.
• "A shift changes the sign." Sometimes. A quarter-turn shift changes the function, which is a different and larger change. Conflating the two is the most productive source of wrong answers in this section.
• "π − A is in the second quadrant, so everything there is negative except the sine." The quadrant rule gives signs and says nothing about which function appears. It is a check on the answer, not a derivation of it.
• "Result 6 needs its own proof." It is result 5 applied to a quarter turn minus the input. Students who miss this treat the pair as two facts and then cannot reproduce either.
• "The sine of a sum is the cosine version with the signs changed." It is not. Result 7 mixes a sine with a cosine in each term, and its derivation goes through results 5 and 4 rather than repeating the geometry of result 3.
• "A full-turn shift is a separate rule." It is the repetition already established on p. 50, arriving a second time through the expansion. Deriving it again is a useful consistency check and nothing more.

Questions to check understanding

• Derive a named entry of result 9 rather than quoting it
• Simplify an expression containing several shifted sines and cosines to a single function
• Prove that a quotient of shifted tangents equals a stated expression
• Produce the tangent or cotangent version of a shift from the sine and cosine versions
• Handle shifts the printed list does not cover, including three quarters of a turn and the eighth-of-a-turn shifts the exercise repeatedly asks for
• Explain why a quarter-turn shift exchanges the two functions and a half-turn shift does not

Examples worth working on the board

Inputs only. Values marked verified are worked out here on the chapter's printed data.

• Result 5 (§3.4, p. 59). The chapter obtains it by putting a quarter turn in place of the first angle and A in place of the second in result 4. Verified: the cosine of a quarter turn is 0 and its sine is 1, so the first product dies and the second survives untouched, leaving the cosine of a quarter turn minus A equal to the sine of A. This is the first time in the chapter that a substitution kills one whole term.
• Result 6 (§3.4, p. 59). Obtained from result 5 by feeding it a quarter turn minus A instead of A. Verified: a quarter turn minus a quarter turn minus A is A again, so result 5 returns the sine of a quarter turn minus A on one side and the cosine of A on the other. Note the shape of the argument: the identity is applied to itself, and no new geometry appears.
• What results 5 and 6 say together (§3.4, p. 59). Verified: two inputs adding to a quarter turn have their sine and cosine interchanged. On the unit circle this is the reflection in the diagonal line through the origin, which exchanges the two coordinates — worth drawing, because it makes both results one picture.
• Result 7 (§3.4, p. 59). The chapter writes the sine of a sum as the cosine of a quarter turn minus that sum — that is result 5 read from right to left, with the whole sum put where its single angle stood, and not result 6, which would need a further rearrangement to reach the same line. It then regroups it as a quarter turn minus A, minus B, and applies result 4. Verified: the two pieces that come out of result 4 are then converted by results 5 and 6, giving the sine of A times the cosine of B plus the cosine of A times the sine of B. Three earlier results, no new geometry.
• Result 8 (§3.4, p. 59). Replace B by −B in result 7. Verified: by results 1 and 2 the cosine of −B is the cosine of B and the sine of −B is minus the sine of B, so the plus becomes a minus and nothing else moves.
• Result 9, the eight entries (§3.4, pp. 59–60). The chapter says these come out of results 3, 4, 7 and 8 by choosing suitable values, and prints them in four pairs. Verified by substitution, and each should be derived rather than displayed: shifting forward by a quarter turn turns the cosine into minus the sine and the sine into the cosine; a half turn less the input leaves the sine alone and reverses the cosine; a half turn more reverses both; a full turn less the input leaves the cosine alone and reverses the sine.
• The mechanism (an added reading of result 9). Verified: every entry is the expansion with one angle set to a quadrantal value, and at such a value one of the two — sine or cosine — is 0 and the other is ±1. At a quarter turn the cosine is the one that dies, so the surviving term carries the other function and the names swap. At a half turn and at a full turn the sine dies, so each surviving term carries its own function and only the ±1 can change anything. That single observation replaces the whole table.
• The tangent's half turn (§3.4, result 9, p. 60). Verified: the half-turn-more pair reverses both the sine and the cosine, so their quotient is unchanged and the tangent of a half turn more than A equals the tangent of A. This is exactly the fact §3.3.2 borrowed on p. 55 to claim that the tangent comes round twice as often as the sine — a genuine forward reference inside this edition, now discharged.
• The page's own extension note (§3.4, p. 60). Immediately after result 9 the chapter records that matching results for the tangent, cotangent, secant and cosecant follow from the sine and cosine versions. Verified: they do, by dividing or inverting the pairs — and the exercises expect them.
• Exercise 3.3, the questions that live on this topic (p. 67). Hand the data over intact. Q6 combines four shifted cosines and sines of an eighth of a turn less two different angles. Q7 is a quotient of two tangents, of an eighth of a turn plus and minus the same angle, to be shown equal to a squared expression in the tangent. Q8 is a quotient of shifted cosines and sines to be shown equal to a squared cotangent. Q9 involves shifts of three quarters of a turn and of a full turn. Q10 uses two consecutive whole-number multiples of an angle. Q11 is a difference of two cosines shifted by three eighths of a turn. Verified, and worth flagging: the shifts these questions need are not the shifts result 9 prints — it lists only a quarter turn forward, a half turn either way and a full turn back, so Q9's three-quarter turn and Q11's three-eighth turn both fall outside it, and they are different shifts, not the same one twice. Note also that Q6, Q7 and Q11 move by eighths of a turn, which is finer than anything in the printed list. All of them have to be produced by substitution into results 3, 4, 7 and 8, exactly as the printed eight were — which is the best possible argument for teaching the method instead of the table.

Figures to have open

• The unit circle with the line through the origin at half a right angle drawn, a point and its reflection marked, and the two coordinate pairs shown exchanging. An added construction; it carries section 4 and the chapter prints no figure for §3.4 beyond Fig 3.14.
• A single substitution frame — the expansion written once with two blank slots — reused for every entry of result 9, so the audience sees the same machine run eight times. An added construction.
• A quadrantal-value strip showing the sine and cosine at 0, a quarter turn, a half turn and three quarters, since these four pairs are the only numbers this topic ever substitutes. Standard schematic.
• The unit circle with a point and its half-turn opposite marked, both coordinates negating, for the tangent argument in section 10. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.4, results 5 and 6, p. 59
• §3.4, results 7 and 8, p. 59
• §3.4, result 9, pp. 59–60, together with the note immediately after it on p. 60 about the remaining four functions
• Exercise 3.3, questions 6 to 11, p. 67
• The chapter Summary, pp. 72–73, reprints results 5 to 9
• Cross-reference inside this edition: §3.3.2, p. 55, announces the tangent's half-turn repetition and defers its proof to this section