Defining sine, cosine and tangent, then their three reciprocals

What to assume they know

• Which side is opposite and which is adjacent depends on the angle you pick — naming the hypotenuse, and naming the opposite and adjacent legs relative to a chosen acute angle
• Writing a ratio of two lengths as a fraction, and reducing it
• What a reciprocal is, and that a reciprocal is undefined when the original is zero
• Dividing one fraction by another, and cancelling a common factor top and bottom
• Reading an abbreviation as a single symbol rather than as a product of letters

What they should be able to do

• Write the definitions of sine, cosine and tangent for a named acute angle of a right triangle in terms of opposite, adjacent and hypotenuse
• State each of cosecant, secant and cotangent as the reciprocal of the correct one of the first three
• Pair each ratio with its reciprocal correctly, and explain why the names do not help
• Derive the relation between tangent, sine and cosine by cancelling the hypotenuse
• Interpret the abbreviations as indivisible symbols and explain why the letters cannot be separated from the angle
• Read and write the squaring convention for these ratios, and distinguish it from the inverse notation
• Produce all six ratios for the other acute angle of the same right triangle
• Recount, in outline, how the word sine travelled from Sanskrit through Arabic and Latin into English

Where it usually goes wrong

• "cosec is short for cosine, so it pairs with secant." It is not and it does not. The reciprocal of sine carries the co- and the reciprocal of cosine does not. Say this out loud in the explanation and show it, because the names are actively misleading and no amount of reasoning will recover them.
• "There are six independent things to learn." There are three, plus a rule for turning them over. A student who can write sine, cosine and tangent can produce the other three on demand.
• "sin A means sin times A." Then sin alone would have a value, and it does not. The three letters are an instruction that only means anything once an angle is supplied.
• "Since tan = sin/cos, tangent is a more complicated ratio." Tangent is the simplest of the three by construction — two legs, no hypotenuse. The relation to sine and cosine is a consequence, not the definition.
• "sin²A and sin A² are the same." The first squares the ratio; the second would square the angle. The convention exists to avoid brackets and it only reads correctly if you know where the exponent belongs.
• "A ratio can be bigger than the sides it came from." These are quotients of lengths, so they are pure numbers with no unit — and two of them, sine and cosine, are bounded, because neither leg can outrun the hypotenuse.
• "Cotangent at R has nothing to do with tangent at P." In a right triangle they are equal, because the two acute angles exchange their legs. Fig. 8.13 makes that a one-line answer rather than a computation.

Questions to check understanding

• Write all six ratios of a named acute angle from a labelled right triangle
• Given one ratio, name its reciprocal and write it
• Simplify an expression by replacing a reciprocal ratio with its partner
• Show that tangent at one acute angle equals cotangent at the other
• True-or-false items on the abbreviations, of the kind Exercise 8.1 question 11 poses
• Short-answer questions on the origin of the word, which the board does ask from the history box

Examples worth working on the board

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

• The definition list (§8.2, p. 115). All six are set out against Fig. 8.4 — triangle ABC, right angle at B, angle A at the lower left. Three are defined directly from the side names, and three are defined as reciprocals of those. In the letters of that figure the six come out as BC/AC, AB/AC, BC/AB, AC/BC, AC/AB and AB/BC.
• The cancellation (§8.2, p. 115). Divide BC/AC by AB/AC. The AC in the two denominators cancels and BC/AB is left. Verified: that is exactly the quotient the chapter calls tangent, so tangent is sine over cosine and, turning the same division around, cotangent is cosine over sine.
• The reciprocal pairing, as a table to build. sine with cosecant, cosine with secant, tangent with cotangent. Worth stressing that three of the six names carry the co- prefix — cosine, cosecant and cotangent — and that the prefix predicts nothing at all about who pairs with whom. Two of the three are matched to a name without the prefix (cosine to secant, cotangent to tangent), and the third, cosecant, is matched to sine. So the prefix marks out a set of three that cuts across the pairing rather than following it.
• A worked instance to make the six concrete (Exercise 8.1 question 1, p. 121). A right triangle with the right angle at B, AB = 24 cm, BC = 7 cm. Verified: AC = 25 cm; at angle A the six ratios are 7/25, 24/25, 7/24, 25/7, 25/24 and 24/7, in the order sine, cosine, tangent, cosecant, secant, cotangent. Each of the last three is visibly one of the first three inverted.
• Fig. 8.13 and the swap (Exercise 8.1 question 2, pp. 121). Triangle PQR with the right angle at Q, PQ = 12 cm and PR = 13 cm printed on the figure — those two numbers appear inside the artwork, not in the question text. The question asks for the tangent at P minus the cotangent at R. Verified: QR = 5 cm, since 169 − 144 = 25. Tangent at P is QR/PQ = 5/12. Cotangent at R is also QR/PQ = 5/12, because at R the leg PQ is the opposite one and QR is the adjacent one. The difference is 0. This is the previous topic's swap doing real work: a tangent at one acute angle and a cotangent at the other are the same number.
• The history box (§8.2, p. 116). Aryabhata, dated on the page C.E. 476–550, used ardha-jya — his name for half of a chord — in the Aryabhatiyam, which the page dates A.D. 500. The word shortened to jya and then jiva, was carried unchanged into Arabic, and was checked into Latin as sinus, a word meaning curve, from which the English sine comes. The abbreviation was introduced by Edmund Gunter, an English professor of astronomy, dated 1581–1626. Aryabhata's name for the cosine idea is given as kotijya; the Latin cosinus is also credited to Gunter, and the short form for cosine to Sir Jonas Moore in 1674. A portrait of Aryabhata sits beside the box.
• The notation warnings (§8.2, pp. 116–117). Two of them, and both are examined. First, the three letters of an abbreviation are one symbol, not a quantity multiplied by the angle — pull the angle away and nothing is left. Second, a squared ratio may be written with the exponent tucked in front of the angle, but a superscript minus one does not mean the reciprocal; it names a different operation the book leaves to later classes.
• True-or-false traps (Exercise 8.1 question 11 items (iii) and (iv), p. 121). One item claims a short form stands for the name of a different ratio; another claims an abbreviation is a product. Verified: both are false, and both are answered by section 8 rather than by any calculation.

Figures to have open

• Fig. 8.4 as a reusable base drawing that the explanation can re-highlight six times — once per ratio — with the numerator side and denominator side picked out in two colours. This is the chapter's own figure; redraw it as a schematic.
• A pairing chart of the six names in two columns, ratio against reciprocal, with all three co- names marked so the false pairing can be shown and then broken. Standard schematic and the single most useful still in this topic.
• Fig. 8.13 redrawn: triangle PQR, right angle at Q, with 12 cm and 13 cm marked where the printed figure marks them, and 5 cm added once it is derived. The chapter prints those two lengths inside the artwork, so anyone working from the question text alone would not have them.
• The Aryabhata portrait from p. 116 is not required. A dated word-chain graphic carries section 10 better than a picture does.

Where this sits in the book

• NCERT Mathematics, Textbook for Class X, Chapter 8 "Introduction to Trigonometry", §8.2 Trigonometric Ratios, p. 115 — the six definitions, the reciprocal note, and the tangent-and-cotangent cancellation
• §8.2, p. 116 — the boxed history of the words, with the Aryabhata portrait, and the Remark on what the abbreviations mean
• §8.2, p. 117 — the Note on the squaring convention, the inverse notation, and the use of the Greek letter for an angle
• Exercise 8.1 questions 1, 2 and 11, p. 121, with Fig. 8.13
• The chapter summary, §8.5, p. 132, points 1 and 2, restates the three definitions and the three reciprocal relations