Coordinates on the unit circle extend the ratios to every real number

What to assume they know

• Why arc divided by radius is the measurement the mathematics prefers — the radian, and radian measures as real numbers
• The exchange rate between the two units, and the arc-length rule it buys — the common angles in both units
• Coordinates in the plane, and that a coordinate can be negative
• Pythagoras' theorem in a right triangle
• The Class X trigonometric ratios of an acute angle, which §3.3 names as its starting point
• The exact ratios at 30°, 45° and 60°, carried forward from Class X

What they should be able to do

• Explain why the ratio definition cannot be applied beyond an acute angle
• Locate the point on the unit circle belonging to a given real number
• State the definition of cosine and sine as coordinates, and check it against the old ratios in the acute case
• Derive the first identity from the equation of the unit circle
• Read off the cosine and sine of the four quarter-turn positions
• Characterise every real number at which sine vanishes, and every one at which cosine vanishes
• Justify why adding any whole number of full turns leaves both values unchanged
• Define the four remaining functions as quotients and state exactly where each fails to be defined
• Derive both of the further identities and name the values each one excludes

Where it usually goes wrong

• "Sine is opposite over hypotenuse." That is a description, not the definition in force from p. 49 onwards. Here sine is the second coordinate of a point. The triangle description is a special case that survives only while the angle is acute.
• "a and b are lengths, so they cannot be negative." They are coordinates. In the second quadrant a is negative, which is exactly how the definition manages to produce a negative cosine without any negative side ever being drawn.
• "cos² + sin² = 1 is a triangle fact." It is the equation of the circle. That is why it needs no restriction, while the other two identities do.
• "One plus tangent squared equals secant squared, always." Not at an odd multiple of a quarter turn, where neither side exists. The chapter's bracketed question is asking for the division that produces the identity, and the division carries the condition with it.
• "tan of a quarter turn is infinity." The table's cell says it is not defined. Infinity is used later, on pp. 53–54, only as a description of how the values behave nearby, and the chapter says so explicitly.
• "Extending the definition changes the familiar values." It does not, and the page says the values at the old angles are unchanged. If a student's new answer for 30° differs from their old one, they have made an error, not a discovery.
• "Quadrantal means in a quadrant." It means on the boundary between two of them — the terminal side lies along an axis, which is why one coordinate is always zero there.

Questions to check understanding

• State the definition of the sine and cosine of a real number and explain why it agrees with the Class X ratios
• Give the values of all six functions at a quadrantal angle, saying which do not exist there
• Solve an equation of the form "sine equals zero" or "cosine equals zero" over the reals, giving the whole family
• Derive one plus the squared tangent equals the squared secant, stating the condition
• State the exclusion set for each of the four quotient functions
• Evaluate a function at an input outside one full turn by first reducing it
• Given one function's value and enough information to fix the sign, find the other five

Examples worth working on the board

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

• Fig 3.6 (§3.3, p. 49). A unit circle drawn on the coordinate axes, centre at the origin O. Four points are lettered on it with their coordinates printed alongside: A at (1, 0) on the right, B at (0, 1) at the top, C at (−1, 0) on the left, D at (0, −1) at the bottom. A fifth point P, lettered with the pair (a, b), sits in the first quadrant; the segment OP is marked 1, the arc from A round to P is marked x, and M is the foot of the perpendicular from P to the horizontal axis, with the piece OM marked a and the piece MP marked b. All the lettering is inside the artwork.
• The definition and the consistency check (§3.3, p. 49). Cosine of x is defined as the first coordinate of P and sine of x as the second. Verified as an argument: in the acute case OMP is a right triangle with hypotenuse OP = 1, so the old "adjacent over hypotenuse" is a/1 and "opposite over hypotenuse" is b/1. The new definition therefore returns exactly the old values wherever the old one applies.
• The first identity (§3.3, p. 49). Pythagoras in triangle OMP gives a² + b² = 1 for every P on the circle, which is the identity between the squared cosine and the squared sine. Verified: this is nothing but the equation of the unit circle, so the identity holds at every real number without exception. The comparison worth making is with the other two identities on p. 51, the ones built from the tangent and the cotangent: those are the ones that fail where a quotient is undefined. Plenty of the chapter's later results are also unconditional — the sum and difference expansions, the tripled-angle pair, the sum-to-product and product-to-sum sets — so this is not the chapter's only unconditional result, only the first, and the one that costs nothing to state.
• The four quarter-turn points (§3.3, pp. 49–50). The chapter records the angles at B, C and D as a quarter, a half and three quarters of a full turn, and calls every whole-number multiple of a quarter turn a quadrantal angle. Reading the coordinates gives ten values at once: at 0 the pair is (1, 0); at a quarter turn (0, 1); at a half turn (−1, 0); at three quarters (0, −1); at a full turn back to (1, 0). Verified: the five listed positions occupy only four distinct points, since the full turn returns to the starting point.
• The two zero sets (§3.3, p. 50). Sine vanishes exactly at the whole-number multiples of a half turn; cosine vanishes exactly at the odd multiples of a quarter turn. Verified: these are the points where P lands on the horizontal and the vertical axis respectively, so the statement is a reading of the picture and not a separate fact.
• Repetition (§3.3, p. 50). Adding any whole number of complete turns lands on the same point of the circle, so neither coordinate moves. Verified: this is why the chapter can later evaluate at inputs far outside one turn, as in Examples 8 and 9 on pp. 56–57.
• The remaining four (§3.3, p. 50). Cosec is one over sine, sec is one over cosine, tan is sine over cosine, cot is cosine over sine. Each carries a printed exclusion: the two built on sine fail at the multiples of a half turn, the two built on cosine fail at the odd multiples of a quarter turn. Verified: the two exclusion sets are exactly the two zero sets from section 7 — no new information is needed to state them.
• The two further identities (§3.3, p. 51). The page states that one plus the squared tangent is the squared secant, and one plus the squared cotangent is the squared cosecant, marking each with a bracketed question rather than a proof. Verified, and this is the answer: divide the first identity by the squared cosine to get the first, and by the squared sine to get the second. The division is only legal where the divisor is non-zero, which is precisely why the first fails at the odd multiples of a quarter turn and the second at the multiples of a half turn.
• The standard-angle table (§3.3, p. 51). Eight columns. Sine runs 0, 1/2, 1/√2, √3/2, 1, 0, −1, 0. Cosine runs 1, √3/2, 1/√2, 1/2, 0, −1, 0, 1. Tangent runs 0, 1/√3, 1, √3, then a cell reading that it is not defined, then 0, then a second not-defined cell, then 0. The page adds that the three reciprocal functions are obtained by inverting these entries. Verified: every tangent entry is the sine entry over the cosine entry, and the two cells that refuse a value fall exactly where the cosine entry is 0 — so the table is self-checking and need not be learnt as eight separate columns. Those two cells are not blank on the page: each carries a printed phrase saying the value does not exist, and a redrawn table must keep that wording rather than leaving a gap.

Figures to have open

• Fig 3.6 redrawn with A, B, C, D, P and M lettered, the radius marked 1, the arc marked x and the two coordinate segments marked. The chapter's own figure, and sections 2 to 8 all read from it. The lettering is inside the printed artwork.
• A step-by-step version of the same circle in which P travels and a coordinate readout updates. An added construction, and the cheapest way to make the definition feel like a function.
• The unit circle with the two zero sets highlighted differently, for section 7. Standard schematic.
• The eight-column standard-angle table, built column by column. This is the chapter's own table on p. 51.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.3 Trigonometric Functions, p. 49 — the framing, Fig 3.6, the definition, and the first identity
• §3.3, p. 50 — the quadrantal angles and their coordinate pairs, the repetition statement, the two zero sets, and the definitions of the four quotient functions with their exclusions
• §3.3, p. 51 — the two further identities with their bracketed questions, and the standard-angle table
• The chapter Summary, p. 72, restates the three identities and the two repetition statements
• Examples 8 and 9, pp. 56–57, are the repetition statement in use and are treated in Where each function is defined, what values it reaches, and how it repeats