Where each function is defined, what values it reaches, and how it repeats

What to assume they know

• Coordinates on the unit circle extend the ratios to every real number — the coordinate definition, the four quotients and their exclusions, and the two zero sets
• Which of the six stays positive where, and what happens at the quarter turns — signs by quadrant, and the reflection identities
• Set-builder notation, and reading a closed interval
• What it means for a function to increase or decrease across a stretch of inputs
• Reciprocals of small numbers, and what happens to 1/t as t nears zero

What they should be able to do

• State the domain of each of the six functions and justify it from the definition
• State the range of each of the six and justify it from the circle or from inverting a bounded set
• Describe how each function rises or falls across each quadrant
• Explain what the infinity symbols in the behaviour table are describing, and what they are not claiming
• State how far apart the repetitions are for each function, and say which of the six differ from the other four
• Identify each of the six graphs from its shape, its vertical breaks and its spacing
• Reduce an input lying outside one turn, or a negative input, before evaluating
• Evaluate any of the six at an input given in degrees or radians well outside the standard range

Where it usually goes wrong

• "All six repeat after a full turn." Four of them do. The tangent and cotangent come back after a half turn, and the chapter says so on p. 55. A student who reduces a tangent by full turns only will get the right answer but do twice the work; a student who reduces a sine by half turns will get the wrong sign.
• "The cosecant's range is the same as the sine's." It is the complement of the open stretch between −1 and 1. Inverting a small number gives a large one, which is exactly why the middle is empty.
• "The tangent has a hole at a quarter turn." A hole would be a single missing point on an otherwise continuous curve. What is there is a vertical break: the values run away in opposite directions on the two sides.
• "Infinity is the value of the tangent at a quarter turn." The chapter's own Remark denies this in as many words. The symbol is shorthand for how the values behave as the input closes in.
• "The graphs are the source of the domain and range." The other way round. The graphs are pictures of facts already established from the circle; drawing them first makes the argument circular.
• "cos(−1710°) is the negative of cos(1710°)." The cosine is unchanged by negating its input — the reflection result of §3.3.1. This trips up more students in Example 9 than the arithmetic does.
• "You need a calculator for the sine at 765°." You need to subtract two full turns. Every one of Exercise 3.2 questions 6 to 10 is that one move.
• "The domain of the tangent excludes the multiples of a half turn." That is the cotangent's exclusion. The tangent loses the odd multiples of a quarter turn. Swapping the two is the most common domain error in the chapter.

Questions to check understanding

• State the domain and range of a named function, with a reason
• Say which of the six repeat after a half turn and why
• Evaluate any of the six at an input given well outside one turn, in degrees or in radians
• Evaluate at a negative input, using the reflection results
• Identify a graph from its shape and the spacing of its breaks
• Explain what the infinity symbol in the behaviour table is asserting
• Decide whether a proposed value, such as a secant of 0.5, is possible at all

Examples worth working on the board

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

• Domains and ranges as printed (§3.3.2, pp. 52–53). Sine and cosine: every real number in, and the closed stretch from −1 to 1 out. Cosecant: every real except the whole-number multiples of a half turn, and the outputs are the numbers at least 1 in size, in either direction. Secant: every real except the odd multiples of a quarter turn, with the same output set. Tangent: every real except the odd multiples of a quarter turn, and every real number as output. Cotangent: every real except the multiples of a half turn, and again every real as output. Verified as the argument: each excluded set is exactly the zero set of that function's denominator, so nothing here is a new fact — it is p. 50 restated.
• Why the two reciprocals abandon the middle. Verified: the sine of anything is at most 1 in size, so its reciprocal is at least 1 in size; and since the sine gets arbitrarily close to zero without reaching it, the reciprocal gets arbitrarily large. Nothing lands strictly between −1 and 1. This is the argument.
• Why tangent and cotangent reach everything. Verified: on the first quadrant the sine climbs from 0 to 1 while the cosine falls from 1 to 0, so their quotient starts at 0 and grows past any bound; by the sign table it then runs over the negatives in the second quadrant. Every real value is hit, which is what the printed range says.
• The behaviour table (§3.3.2, p. 53). Six rows, four columns, one entry per quadrant. Sine: up 0 to 1, down 1 to 0, down 0 to −1, up −1 to 0. Cosine: down 1 to 0, down 0 to −1, up −1 to 0, up 0 to 1. Tangent: up from 0 without bound, up from below without bound to 0, up from 0 without bound, up from below without bound to 0. Cotangent: down without bound to 0, down from 0 without bound, down without bound to 0, down from 0 without bound. Secant: up 1 without bound, up from below without bound to −1, down from −1 without bound, down without bound to 1. Cosecant: down without bound to 1, up 1 without bound, up from below without bound to −1, down from −1 without bound. Verified as structure: the cosine row is the sine row shifted one quadrant, and the four rows below are forced by the two above them together with the sign table.
• The Remark on infinity (§3.3.2, pp. 53–54). The chapter states plainly that the two infinity symbols are describing a kind of behaviour and are not values. Hand this over intact — it is the single most useful sentence on the page for heading off the misconception below.
• Repetition (§3.3.2, pp. 54–55). Sine and cosine come round after a full turn, and therefore so do cosecant and secant. Tangent comes round after a half turn, and so does cotangent because it is the tangent's reciprocal. The chapter is explicit that the tangent result is being taken on credit from §3.4, where it is proved. Verified: on the unit circle, adding a half turn sends the point to its opposite, negating both coordinates, so their quotient is unchanged — that is the one-line reason, and it can be shown on the circle before §3.4 arrives.
• The six graphs (§3.3.2, pp. 54–55). Fig 3.8 is the sine, Fig 3.9 the cosine, Fig 3.10 the tangent, Fig 3.11 the cotangent, Fig 3.12 the secant, Fig 3.13 the cosecant; each carries its equation as a caption beneath the curve. The sine and cosine are drawn from −4π to 4π: two full turns each side of the origin, four in all, with the horizontal axis marked at whole multiples of π and the vertical at 1 and −1. The other four are drawn over a narrower stretch, roughly from −π to 2π, with the vertical breaks shown as dashed lines and the vertical axis marked at 1, 2, −1 and −2. All of this is read off pp. 54 and 55.
• Example 8 (§3.3, p. 56). Evaluate the sine at 31π/3. Inputs only. Verified: 31π/3 is π/3 plus five full turns, so the value is the sine of π/3, namely √3/2.
• Example 9 (§3.3, p. 57). Evaluate the cosine at −1710°. Inputs only. Verified: adding five full turns of 360° gives 90°, so the value is 0.
• Exercise 3.2, questions 6 to 10 (p. 57). Hand the data over intact: the sine at 765°; the cosecant at −1410°; the tangent at 19π/3; the sine at −11π/3; the cotangent at −15π/4. Verified, as working added here: 765° is 45° past two full turns, giving 1/√2. −1410° plus four full turns is 30°, so the cosecant is 2. 19π/3 is π/3 past three full turns, giving √3 — and note the tangent would have come home after a half turn, so this one can be reduced faster than the others. −11π/3 plus two full turns is π/3, giving √3/2. −15π/4 plus two full turns is π/4, giving 1. Every item is testing the same reduction, and three of the five — Q7, Q9 and Q10 — are testing whether a student can add turns to a negative input without losing the sign. They are not the last three: Q8's input is a positive multiple of π/3 and needs no sign handling at all, which is what makes it the odd one out in the middle of the set.

Figures to have open

• Figs 3.8 and 3.9 redrawn on one axis so the quarter-turn offset between the two curves is visible. The chapter draws them separately across −4π to 4π; the comparison is added here.
• Figs 3.10 and 3.11 redrawn with the vertical breaks dashed and the repeat distance marked as a half turn, beside the sine's full turn. The chapter's own figures.
• Figs 3.12 and 3.13 redrawn with the corresponding cosine and sine drawn faintly underneath, so each break lines up with a zero. The chapter prints them without that overlay, and the overlay is the point of section 4.
• The unit circle with a point and its opposite marked, both coordinates negated, for the half-turn argument in section 9. An added construction.
• A number line with the excluded inputs of each function punched out, in six parallel strips. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.3.2 Domain and range of trigonometric functions, pp. 52–53 — the six domains, the six ranges, and the behaviour table
• §3.3.2, pp. 53–54 — the Remark on the two infinity symbols
• §3.3.2, pp. 54–55 — the repetition statements and Figs 3.8 to 3.13
• §3.3, pp. 56–57 — Examples 8 and 9
• Exercise 3.2, questions 6 to 10, p. 57
• Cross-reference inside this edition: the tangent's half-turn repetition is stated on p. 55 and justified by result 9 of §3.4, p. 60