The exchange rate between the two units, and the arc-length rule it buys

What to assume they know

• An angle as an amount of turning, and what its sign records — degree measure, and minutes and seconds as sixtieths
• Why arc divided by radius is the measurement the mathematics prefers — the radian, and the arc-length relation in radians
• Solving a proportion for any one of its four entries
• Handling π both as a symbol and as the working values 22/7 and 3.14
• Improper fractions and mixed numbers, since a degree-minute measure is one

What they should be able to do

• Derive both conversion multipliers from the single half-turn equation
• Convert a degree measure, including minutes, into radians
• Convert a radian measure into degrees, minutes and seconds
• Reproduce the table of common angles in both units and use it as a check
• Apply the chapter's notational convention: read a bare number as a radian measure
• Find any one of arc, radius and angle from the other two
• Convert a rotation rate into an angle turned and then into a distance travelled
• Explain why, for one fixed arc length, the radii of two circles vary inversely with the angles the arc subtends in them

Where it usually goes wrong

• "To convert, multiply by π/180." Only in one direction. The reliable habit is to write the half-turn equation down first and divide it the way the question needs; the multiplier then cannot be inverted by accident.
• "40° 20′ is 40.20°." It is 40⅓°, because the prime mark is a sixtieth. Example 1 turns on this and nothing else.
• "22/7 and 3.14 are both π, so it does not matter which you use." The chapter names the value to use in Examples 2, 3 and 4 and in Exercise 3.1 questions 2 and 4, and the last digit of the answer moves with the choice. The value used is part of the answer.
• "l = rθ works in degrees if you are careful." It does not work in degrees at all. Every one of Examples 3 and 4 and Exercise 3.1 questions 4, 5 and 7 begins by converting, and that step is the exercise.
• "6.28 cm is how far the tip ends up from where it started." It is the length of the path travelled. After 40 minutes the tip is 1.5√3 cm from its starting point in a straight line, which is nothing like 6.28.
• "Bigger angle, bigger circle." For one fixed arc, the opposite: the circle that wraps it into the larger angle is the smaller circle. Example 5 and Exercise 3.1 question 6 both make that point, and students routinely write the ratio the wrong way up.
• "The chord in Exercise 3.1 question 5 needs trigonometry." It needs the observation that the chord is as long as the radius. The triangle is equilateral and the angle falls out with no ratios at all.

Questions to check understanding

• Convert a degree-minute measure into radians and a radian measure into degrees, minutes and seconds
• State the radian equivalent of a common angle without working
• Find the angle in degrees subtended at the centre by a given arc in a circle of given radius
• Convert revolutions per minute into radians per second
• Find the minor arc cut off by a chord of stated length in a circle of stated diameter
• Given equal arcs in two circles and the two angles, find the ratio of the radii
• Find the angle swept by a pendulum tip from the length and the arc traced

Examples worth working on the board

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

• The founding equation (§3.2.4, p. 46). A full turn is 2π radian and also 360°, so a half turn is π radian and 180°. Verified: dividing that both ways gives a radian-per-degree multiplier of π/180 and a degree-per-radian multiplier of 180/π, and the two multiply to 1 — which is the check to run whenever a student is unsure which way up to write it.
• The size of each unit (§3.2.4, p. 46, working with π taken as 22/7). The page reports one radian as about 57°16′ and one degree as about 0.01746 radian. Verified: 180 ÷ (22/7) = 1260/22 = 57.2727…, whose fractional part times 60 is 16.36, giving 57°16′; and (22/7) ÷ 180 = 22/1260 = 0.017460…. Note when explaining it that with the more accurate π the first value is 57°17′45″, so the chapter's minute figure depends on its own choice of π — a good moment to make the point that 22/7 is a working value, not π.
• The common-angle table (§3.2.4, p. 46). Seven columns, in this order: 30° with π/6, 45° with π/4, 60° with π/3, 90° with π/2, 180° with π, 270° with 3π/2, 360° with 2π. Hand the pairs over intact. Verified as a self-check: each radian entry is 180° divided by the number under π, and the last three are the half, three-quarter and full turns, so the table can be rebuilt rather than memorised.
• Notational Convention (p. 47). A degree sign written on an angle means the degree measure is that number; a plain symbol with no sign means the radian measure is that number, and the word itself is usually left off. So the printed π and π/4 are 180° and 45°.
• Example 1 (p. 47). Convert 40° 20′ into radians. Inputs only: 40 degrees and 20 minutes. Verified: 20 minutes is a third of a degree, so the measure is 121/3 degrees; multiplying by π/180 gives 121π/540 radian.
• Example 2 (p. 47). Convert 6 radians into degrees, with π taken as 22/7. Verified: 6 × 180 × 7 ÷ 22 = 7560/22 = 343 7/11 degrees; 7/11 of a degree is 38 2/11 minutes; 2/11 of a minute is about 10.9 seconds, so the result is about 343° 38′ 11″. The point of the example is the two successive sixtieth conversions.
• Example 3 (pp. 47–48). Arc length 37.4 cm, central angle 60°, π taken as 22/7; find the radius. Inputs only. Verified: 60° is π/3 radian, so the radius is 37.4 × 3 ÷ π = 112.2 × 7 ÷ 22 = 35.7 cm.
• Example 4 (p. 48). A watch, its minute hand measuring 1.5 cm, and a stretch of 40 minutes; the question asks for the distance covered by the far end of that hand, with π taken as 3.14. Inputs only. Verified: 40 minutes is two-thirds of a revolution, so the angle turned is 240°, which is 4π/3 radian, and the distance is 1.5 × 4π/3 = 2π ≈ 6.28 cm. Two things worth saying: the answer is a path length, not a displacement, and a real minute hand turns clockwise, so under the chapter's own sign rule the angle is negative and the example is quietly using its size only.
• Example 5 (p. 48). Two circles, arcs of equal length, subtending 65° and 110° at the respective centres; find the ratio of the radii. Inputs only. Verified: equal arcs give r₁θ₁ = r₂θ₂, so the radii are in the ratio 110 : 65, that is 22 : 13 — and note the conversion to radians cancels out entirely, so the ratio could have been read straight off the degrees.
• Exercise 3.1 (pp. 48–49). Hand the data over intact. Q1, degrees into radians: 25°, −47° 30′, 240°, 520°. Q2, radians into degrees with π as 22/7: 11/16, −4, 5π/3, 7π/6. Q3: a wheel turning 360 revolutions in one minute; radians per second. Q4: radius 100 cm, arc 22 cm, π as 22/7; the angle in degrees. Q5: diameter 40 cm, a chord of length 20 cm; the minor arc it cuts off. Q6: equal arcs subtending 60° and 75°; the ratio of the radii. Q7: a pendulum of length 75 cm whose tip traces arcs of 10 cm, 15 cm and 21 cm; the angle in radians in each case. Verified, as working added here: Q1 gives 5π/36, −19π/72, 4π/3 and 26π/9. Q2 gives 39° 22′ 30″, about −229° 5′ 27″, 300° and 210°. Q3 gives 12π. Q4 gives 12° 36′. Q5 is the one with a hidden step — the chord equals the radius, so the triangle is equilateral and the angle is π/3, making the arc 20π/3 cm. Q6 gives 5 : 4. Q7 gives 2/15, 1/5 and 7/25 radian.

Figures to have open

• The half-turn equation as a single annotated diagram, with both divisions shown as branches from it. Standard schematic.
• The seven-column common-angle table, built rather than displayed. This is the chapter's own table on p. 46.
• A watch face with a 1.5 cm minute hand, the swept sector shaded and the traced arc drawn thick, for Example 4. Standard schematic; the chapter prints no figure for this example.
• Two circles of different radii with an arc of the same length marked on each, for Example 5 and Exercise 3.1 question 6. Standard schematic; the chapter prints no figure for either.
• A circle of diameter 40 cm with a 20 cm chord and the equilateral triangle it forms with the two radii, for Exercise 3.1 question 5. Standard schematic.

Where this sits in the book

• NCERT Mathematics, Textbook for Class XI, Chapter 3 "Trigonometric Functions", §3.2.4 Relation between degree and radian, p. 46 — the founding equation, both approximate unit sizes, and the common-angle table
• The unnumbered heading Notational Convention, p. 47, with the two conversion statements set out beneath it
• Examples 1 and 2, p. 47; Example 3, pp. 47–48; Examples 4 and 5, p. 48
• Exercise 3.1, questions 1 to 7, pp. 48–49
• The first three items of the chapter Summary, p. 72, restate the arc relation and both conversions