An angle as an amount of turning, and what its sign records

What to assume they know

• Rays, endpoints, and the ordinary corner-angle of Classes VI–X
• The trigonometric ratios of an acute angle as side quotients in a right triangle, which §3.1 explicitly takes as the starting point
• Reading a fraction of a whole, since a degree is defined as a fraction of a complete turn
• Place value and the idea of a base other than ten, for the sixtieths that make minutes and seconds

What they should be able to do

• State what a rotation-based angle records that a corner-angle cannot
• Name the initial side, the terminal side and the vertex on a drawn angle
• Decide the sign of a drawn angle from its direction of turn alone
• Explain why one complete revolution is a convenient unit for fast-spinning objects and an inconvenient one for small angles
• Say what fraction of a revolution one degree is, and why 360 was a workable choice
• Convert between a degree-minute-second measure and a fraction of a degree
• Read every panel of Fig 3.3 and say how many complete turns each angle contains
• Explain why the terminal side alone does not determine the angle

Where it usually goes wrong

• "An angle cannot be more than 360°." That ceiling belonged to the corner definition. Under the rotation definition, 420° is one full turn and a bit more, and the chapter draws it on p. 44.
• "A negative angle is somehow less than nothing." The minus sign is a direction flag, not a size. −30° and 30° are the same amount of turning done opposite ways, which is exactly what the two panels of Fig 3.1 show.
• "The picture tells you the angle." It does not. A drawn terminal side is consistent with infinitely many angles differing by whole turns.
• "40° 20′ means 40.2°." The prime is a sixtieth, so it means 40⅓°. Feeding the decimal reading into the conversions of §3.2.4 gives a wrong answer that looks plausible.
• "Revolutions are a childish unit." They are the unit the definition suggests first, and the right one for a wheel at 15 turns a second. Units are chosen for the size of the thing measured.
• "Trigonometry is about triangles." It started there, and the chapter's own opening list is about waves, circuits and vibration. What changes is the role the triangle plays: it stops being the definition, and the unit circle takes over. It does not disappear — §3.3 on p. 49 draws a right triangle inside that circle and gets the first identity out of Pythagoras, the argument on p. 58 turns on two triangles being congruent, and the Historical Note on p. 75 closes with similar triangles.

Questions to check understanding

• Given a drawn rotation, name the initial side, the terminal side and the vertex, and state the sign
• Convert a degree-minute-second measure into a fraction or decimal of a degree and back
• Express a rate given in revolutions per second as degrees per second
• Given an angle beyond one turn, state how many complete revolutions it contains and what is left over
• Give two different angles with the same terminal side, and say what they differ by
• Decide whether a stated measure is positive or negative from a description of the turning

Examples worth working on the board

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

• The opening survey (§3.1, p. 43). The chapter lists present-day users of trigonometry: seismology, the design of electric circuits, the description of atomic states, the prediction of ocean tide heights, and the analysis of a musical tone. Hand the list over intact — the point of section 1 is that the subject outgrew triangles, and the list is the evidence. The page also carries a portrait captioned with the name Arya Bhatt and the dates 476–550.
• Fig 3.1, two panels (§3.2, pp. 43–44). Left panel: vertex O at the lower left, the initial side running right to A, the terminal side running up-right to B, captioned as the positive case. Right panel: vertex O at the upper left, the initial side running right to A, the terminal side running down-right to B, captioned as the negative case. Both captions sit inside the artwork. Note that the two panels differ only in which way the arrow sweeps — the drawn corner is about the same size in each.
• Fig 3.2 (§3.2, p. 44). A single small diagram: the initial side to A, the terminal side to B lying almost on top of it, and a complete loop drawn at O. This is the picture of one revolution — the finishing ray is back where it started and the angle is nevertheless a whole turn.
• The spinning wheel (§3.2, p. 44). The chapter's own figure for why revolutions are a sensible unit: a wheel turning at 15 revolutions per second. Verified: that is 5400 degrees per second, so a sixtieth of a second already carries 90 degrees — a right angle in the blink of an eye, which makes the case for the coarser unit better than any argument.
• The definition of a degree (§3.2.1, p. 44). One degree is the angle of a rotation equal to one three-hundred-and-sixtieth of a revolution. Verified consequences to show: a right angle is 90 of them, a straight angle 180, and 360 divides exactly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180 — twenty-two divisors besides 1 and the number itself, twenty-four counting those two, which is why the number survives.
• Sixtieths (§3.2.1, p. 44). One degree is 60 minutes and one minute is 60 seconds. Verified: one degree is therefore 3600 seconds, and the measure 40° 20′ used later in the chapter equals 40⅓ degrees, not 40.2 degrees — the chapter itself takes exactly that step at the top of p. 47.
• Fig 3.3, six panels (§3.2.1, p. 44). The angles drawn are 360°, 180°, 270°, 420°, −30° and −420°, in that order, three across the top row and three across the bottom. Read off the printed page: the 360° panel shows the terminal side lying on the initial side with one loop at O; the 420° panel shows one loop plus a terminal side raised above the initial side by something well short of a right angle — measured on the panel, the drawn ray sits at about 49°, so the artwork is loose about the 60° the arithmetic requires and an explanation should not invite anyone to read the size off the picture; the −30° panel shows the terminal side just below the initial side with no loop; the −420° panel shows a clockwise loop plus a terminal side well below the initial side. Verified: 420° = 360° + 60°, and −420° = −(360° + 60°), so its terminal side sits 60° below the initial side — the same finishing position a rotation of 300° would produce.
• The turnover to make explicit. Verified: the 360° panel and a rotation of 0° leave the ray in identical positions, and the 420° panel and a rotation of 60° do too. The picture of the finished ray is the same; the angle is not.

Figures to have open

• Fig 3.1 redrawn as two panels sharing one vertex and one initial side, with the sweep shown step by step in opposite directions. The chapter's own figure; the words identifying the two panels are inside the artwork and must be reproduced as labels.
• Fig 3.2 redrawn: one closed loop at the vertex with the terminal side settling back onto the initial side.
• Fig 3.3 redrawn as six panels in the printed order — 360°, 180°, 270°, 420°, −30°, −420° — with the number of complete loops shown as a separate counter beside each. This is the chapter's own figure and section 9 cannot be taught without it.
• A clock-face style dial marked in degrees, minutes and seconds for section 8. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.1 Introduction, p. 43 — the etymology, the application list, and the statement that the chapter generalises ratios into functions
• §3.2 Angles, pp. 43–44 — the rotation definition, Fig 3.1, Fig 3.2, the sign convention, and the revolution as a unit
• §3.2.1 Degree measure, p. 44 — the definition of a degree, minutes and seconds, and Fig 3.3
• The chapter's Historical Note, pp. 74–75, gives the same mathematician's name a second spelling, Aryabhatta, and dates him 476 only