One distance calculation on the unit circle yields the cosine of a difference

What to assume they know

• Coordinates on the unit circle extend the ratios to every real number — cosine and sine as coordinates on the unit circle, and the identity between their squares
• Which of the six stays positive where, and what happens at the quarter turns — the effect of negating the input, which §3.4 lists as its results 1 and 2
• The distance formula between two points given by coordinates
• Congruence of triangles by two sides and the included angle
• Expanding a squared bracket and collecting like terms

What they should be able to do

• State what §3.4 is allowed to assume before result 3 is proved
• Place four points on the unit circle from their arcs and write down their coordinates
• Justify the congruence the chapter marks with a bracketed question
• Expand each squared chord and use the first identity to collapse it
• Complete the proof of result 3 and say where each cancellation came from
• Explain why the squared distance between two points of the unit circle depends only on the difference of their arcs
• Obtain result 4 from result 3 by negating one input, naming the two results that make the step legal
• Explain why no further geometric argument is needed anywhere in §3.4

Where it usually goes wrong

• "The cosine of a sum is the sum of the cosines." The single most common error in the chapter, and the reason §3.4 exists. Kill it with the quarter-turn counterexample before anything else is said.
• "The two chords are equal because the picture looks symmetric." They are equal because two sides and an included angle match. The chapter's bracketed question is asking for exactly that, and leaving it unanswered leaves the proof with a hole.
• "The 2 appears because there are two points." It appears because the first identity was used twice. Every 2 in the derivation is traceable, and tracing them is what makes the proof reproducible in an examination.
• "The difference formula has to be proved separately." It is result 3 with the second input negated. If a student cannot do that substitution, they will treat the next seventeen results as seventeen things to memorise.
• "The proof only works for the angles drawn." The algebra is coordinate algebra and holds for any real inputs; but the chapter's congruence argument is drawn for one arrangement of the four points, and a careful student is right to notice that. Say so rather than pretending the figure is general.
• "Sine and cosine of a sum are related the same way, with a plus sign." They are not. Result 7 has a different shape entirely, and guessing it from result 3 produces a formula that fails the first numerical check.

Questions to check understanding

• Prove the cosine of a sum from the unit circle, including the congruence step
• Justify the collapse of the four squared terms to 2
• Derive the cosine of a difference from the cosine of a sum
• Evaluate the cosine of 15° or 75° by splitting the angle into two standard ones
• Show that a stated expression in two angles simplifies to the cosine of their sum or difference
• Explain why the squared chord between two points of the unit circle depends only on their arc difference

Examples worth working on the board

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

• The starting stock (§3.4, p. 57). The section opens by listing as results 1 and 2 the two facts already obtained on p. 51: negating the input reverses the sine and leaves the cosine alone. Those two, the coordinate definition, the first identity and the distance formula are the entire toolkit for result 3.
• The guess worth killing first. Verified as a counterexample to show: if the cosine of a sum were the sum of the cosines, then taking both angles to be a quarter turn would give a cosine of a half turn equal to 0 + 0 = 0. The true value is −1. One line, and the need for a genuine theorem is established.
• Fig 3.14 (§3.4, p. 58). The unit circle on the coordinate axes, centre O, with four points lettered P₁, P₂, P₃ and P₄. P₄ sits at (1, 0) on the right and its coordinates are printed beside it. P₁ is drawn in the upper left with its coordinate pair printed as the cosine and sine of A; P₂ is drawn in the lower left with the cosine and sine of A + B; P₃ is drawn below the centre and to the left of the vertical axis, with the cosine and sine of −B — the drawn second angle is big enough that its negation lands in the lower-left region, not the lower-right one a small angle would suggest. Measured on the printed page, P₃ sits directly below P₁, on the same vertical line. Sweeping arcs are drawn outside the circle and lettered with the angles A, B and −B. Everything named here is lettering inside the artwork, read off p. 58.
• Why the coordinates need no work (§3.4, p. 58). Each point sits at a stated arc from P₄, so by the definition of p. 49 its coordinates are the cosine and the sine of that arc. This is the step where the definition earns its keep: had sine and cosine still been triangle ratios, three of these four points could not have been written down.
• The congruence, and the first bracketed question (§3.4, p. 58). The chapter asserts that triangle P₁OP₃ is congruent to triangle P₂OP₄, and prints a bracketed question in place of the reason. Verified — this is the answer: all four of OP₁, OP₂, OP₃ and OP₄ are radii of the unit circle and so equal 1; the angle at O in the first triangle runs from P₁ back to P₄ and on to P₃, which is A + B; the angle at O in the second runs from P₂ to P₄, which is also A + B. Two sides and the included angle match, so the triangles are congruent and the chords P₁P₃ and P₂P₄ have the same length.
• The first chord squared, and the second bracketed question (§3.4, p. 58). Apply the distance formula to P₁ and P₃ and expand. Verified: the expansion produces the squared cosine and squared sine of A, the squared cosine and squared sine of B, and two cross terms. The chapter then writes 2 in place of the four squares and attaches a bracketed question; the answer is that the first identity is being used twice, once on A and once on B. What is left is 2 minus twice the quantity (cosine A times cosine B, minus sine A times sine B).
• The second chord squared (§3.4, p. 58). Apply the distance formula to P₂ and P₄ and expand. Verified: the squared cosine and squared sine of A + B again sum to 1 by the first identity, leaving 2 minus twice the cosine of A + B.
• Result 3, obtained (§3.4, pp. 58–59). Equate the two expressions. The 2 on each side cancels and so does the factor of 2, leaving the cosine of A + B equal to the product of the two cosines minus the product of the two sines. Verified: no step in the chain used anything about the sizes of A and B beyond the configuration drawn.
• What was really measured. Verified, and worth a section of its own: run the same distance computation on two arbitrary points of the unit circle, at arcs u and v. The squared distance comes out as 2 minus twice the quantity (cosine u times cosine v plus sine u times sine v), which by result 4 is 2 minus twice the cosine of u − v. So the chord length is a function of the arc difference alone — which is exactly why the chapter's four points, chosen so that two different pairs have the same separation, produce the identity.
• Result 4 (§3.4, p. 59). Replace B by −B in result 3. Verified: results 1 and 2 turn the cosine of −B back into the cosine of B and the sine of −B into minus the sine of B, so the subtraction becomes an addition and the cosine of A − B is the product of the cosines plus the product of the sines. No new geometry is involved.
• A numerical check to run. Verified: take A as a third of a half turn and B as a sixth of one, that is 60° and 30°. Result 3 gives the cosine of 90° as (1/2)(√3/2) − (√3/2)(1/2) = 0, and result 4 gives the cosine of 30° as (1/2)(√3/2) + (√3/2)(1/2) = √3/2. Both agree with the table on p. 51.

Figures to have open

• Fig 3.14 redrawn with the four points, their coordinate pairs and the three lettered arcs. The chapter's own figure, and sections 3 to 8 read directly from it; the lettering is inside the printed artwork.
• The same circle with the two triangles separated out and their equal parts marked. An added construction, and the only way to make the congruence argument visible.
• A movement of two points sliding round the circle with their separation held fixed and the chord length displayed. An added construction, carrying section 9.
• A side-by-side of results 3 and 4 with the two sign changes traced back to results 1 and 2. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.4 Trigonometric Functions of Sum and Difference of Two Angles, p. 57 — the framing and results 1 and 2
• §3.4, result 3, pp. 58–59, with Fig 3.14 on p. 58 — the congruence, both distance computations and the conclusion
• §3.4, result 4, p. 59 — the difference form
• Both results are restated in the chapter Summary, p. 72
• Exercise 3.3 question 6, p. 67, and Miscellaneous Exercise questions 3 and 4, pp. 71–72, are these two results in use