Expanding the cosine and sine by Maclaurin's theorem, we have the usual series,—

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cos z = 1 − z²/2! + z⁴/4!-z⁶/6!+z⁸/8! − &c.

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sin z = z − z⁸/3!+z⁵/5!-z⁷/7!+z⁹/9! − &c.

where z is expressed in radians.

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These series are not in convenient form for numerical calculation, so put z = m/n · π/2

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then cos z = 1 − m²/n² · ½!(π/2)² + m⁴/n⁴ · ¼!(π/2)⁴ − m⁶/n⁶ · ⅙!(π/2)⁶+&c.

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and sin z = m/n · π/2-m⁸/n⁸ · ⅓!(π/2)⁸ + m⁵/n⁵ · ⅕!(π/2)⁵-&c.

For convenience these may be written

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cos z = 1 − m²/n² · A + m⁴/n⁴ · B − m⁶/n⁶ · C + m⁸/n⁸ · D − &c. (P)

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and sin z = m/n · a − m³/n³ · b + m⁵n⁵ · c − m⁷/n⁷ · d + &c. (Q)

where (Ćallet, pages 27-and 28)—

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A = ½! · (π/2)² = 1·23370, 05501, 36169, 82735, 43

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B = ¼! · (π/2)⁴ = 0·25366, 95079, 01048, 01363, 66

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C = ⅙! · (π/2)⁶ = 0·02086, 34807, 63352, 96087, 31

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D = ⅛! · (π/2)⁸ = 0·00091, 92602, 74839, 42658, 02

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E = 1/10! · (π/2)¹⁰ = 0·00002, 52020, 42373, 06060, 55

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F = 1/12! · (π/2)¹² = 0·00000, 04710, 87477, 88181, 72

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G = 1/14! · (π/2)¹⁴ = 0·00000, 00063, 86603, 08379, 19

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H = 1/16! · (π/2)¹⁶ = 0·00000, 00000, 65659, 63114, 98

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I = 1/18! · (π/2)¹⁸ = 0·00000, 00000, 00529, 44002, 01

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J = ½0! · (π/2)²⁰ = 0·00000, 00000, 00003, 43773, 92

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K = ½2! · (π/2)²² = 0·00000, 00000, 00000, 00008, 21

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M = ½6! · (π/2)²⁶ = 0·00000, 00000, 00000, 00000, 03 &c., and

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a = π/2 = 1·57079, 63267, 94896, 61923, 13

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b = ⅓! · (π/2)³ = 0·64596, 40975, 06246, 25365, 58

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c = ⅕! · (π/2)⁵ = 0·07969, 26262, 46167, 04512, 05

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d = 1/7! · (π/2)⁷ = 0·00468, 17541, 35318, 68810, 07

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e = 1/9! · (π/2)⁹ = 0·00016, 04411, 84787, 35982, 19

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f = 1/11! · (π/2)¹¹ = 0·00000, 35988, 43235, 21208, 53

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g = 1/13! · (π/2)¹³ = 0·00000, 00569, 21729, 21967, 93

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h = 1/15! · (π/2)¹⁵ = 0·00000, 00006, 68803, 51098, 11

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i = 1/17! · (π/2)¹⁷ = 0·00000, 00000, 06066, 93573, 11

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j = 1/19! · (π/2)¹⁹ = 0·00000, 00000, 00043, 77065, 47

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k = 1/21! · (π/2)²¹ = 0·00000, 00000, 00000, 25714, 23

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l = 1/23! · (π/2)²³ = 0·00000, 00000, 00000, 00125, 39

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m = 1/25! · (π/2)²⁵ = 0·00000, 00000, 00000, 00000, 52, &c.

By means of these formulæ the sines and cosines of angles are readily obtained: and the calculation of the leading differences for the formation of a table of natural sines is described by Callet thus:—

“S'il est question de trouver les sinus d'une suite d'arcs en progression arithmétique; on peut à l'aide du calcul des differences finies; tirer des formules précédentes, d'autres formules qui donnent les différences premieres, secondes, troisiemes, &c., de ces quantités: pour cela, reprenons la formule Q

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sin m/n · π/2 = m/n · a − m⁸/n⁸ · b + m⁵/n⁵ · c − m⁷/n⁷ · d + &c.

Substituons, dans cette équation Q,m + Δm à m; il viendra une équation Q¹ de laquelle ôtant l'équation Q, nous aurons

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Δ sin m/n · π/2 = a · Δm/n

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− b · Δm/n⁸(3m² + 3mΔm + δm²) + c · Δm/n⁵(5m⁴ + 10m⁸δm + 10m²δm² + 5mδm⁸ + δm⁴)

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− d · Δm/n⁷ (7m⁶+21m⁵Δm + 35m⁴Δm² + 35m⁸Δm⁸ + &c.)

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+ e · Δm/n⁹ (9m⁸ + 36m⁷Δm + 84m⁶Δm² + &c.)

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− f · Δm/n¹¹ (11m¹⁰ + 55m⁹Δm + &c.)

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+ g · Δm/n¹³ (13m¹² + &c.) (ΔQ)

“Nous trouverons de měme en faisant Δm constant

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Δ² sin m/n · π/2 = − b · Δm²/n³ (6m + 6δm)

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+ c · Δm²/n⁵ (20m³ + 60m²Δm + 70mΔm² + 30Δm³)

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− d · Δm²/n⁷ (42m⁵ + 210m⁴Δm + 490m³Δm² + 630Δm²Δm³ + 434m · Δm⁴ + 126Δm⁵)

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+ e · Δm²/n⁷ (72m⁷ + 504m⁶Δm + 1764m⁵ Δm² + 3780m⁴Δm³ + &c.)

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− f · Δm²/n¹¹ (110m⁹ + 990m⁸Δm + 4620m⁷ Δm² + &c.)

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− g · Δm²/n¹⁸ (156m¹¹ + 171m¹⁰ Δm + &c.)

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− h · Δm²/n¹⁵ (210m¹³ + &c.) + &c. (Δ²Q

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Δ³ sin m/n · π/2 = − 6b · Δm³/n³ + c · Δm³/n⁵ (60m² + 180mΔm + 150Δm²)

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− d · Δm³/n⁷ (210m⁴ + 1260m⁸Δm + 3150m²Δm² + 3780ΔmΔm³ + 1806 Δm⁴)

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+ e · Δm³/n⁹ (504m⁶ + 4536m⁵Δm + 18900m⁴Δm² + 45360m³Δm³ + &c.)

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− f · Δm⁸/n¹¹ (990m⁸ + 11880m⁷Δm + 69300m⁶Δm² + &c.)

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+ g · Δm⁸/n¹³ (1716m¹⁰ + 25740m⁹Δm + 193050m⁸Δm² + &c.)

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− h · Δm³/n¹⁵ (2730m¹² + 49140m¹¹Δm + &c.) + &c. (Δ⁸Q)

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Δ⁴ sin m/n π/2 = c · Δm⁴/n⁵ (120m + 240Δm)

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− d · Δm⁴/n⁷ (840m³ + 5040m²Δm + 10920m² + 8400Δm⁸)

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+ e · Δm⁴/n⁹ (3024m⁵ + 30240m⁴Δm + 131040m³Δm² + &c.)

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− f · Δm⁴/n¹¹ (7920m⁷ + 110880m⁶Δm + &c.)

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+ g · Δm⁴/n¹³ (17160m⁹ + 308880m⁸Δm + &c.)

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− h · Δm⁴/n¹⁵ (32760m¹¹ + &c.) + &c. (Δ⁴Q)

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Δ⁵ m/n π/2 = 120c · Δm⁵/n⁵ − d · Δm⁵/n⁷ (2520m² + 12600mΔm + 16800Δm²)

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+ e · Δm⁵/n⁹ (15120m⁴ + 151200m³Δm + 604800m²Δm² + &c.)

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− f · Δm⁵/n¹¹ (55440m⁶ + 831600m⁵Δm + 5544000m⁴Δm² + &c.)

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+ g · Δm⁵/n¹³ (154440m⁸ + 3088800m⁷Δm + &c.)

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− h · Δm⁵/n¹⁵ (360360m¹⁰ + &c.) + &c. (Δ5Q)

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Δ⁶ sin m/n π/2 = − d · Δm⁶/n⁷ (5040m + 15120Δm)

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+ e · Δm⁶/n⁹ (60480m³ + 544320m²Δm + 1723680mΔm² + 1905120Δm³)

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− f · Δm⁶/n¹¹ (332640m⁵ + 4989600m⁴Δm + 31600800m³Δm² + &c.)

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+ g · Δm⁶/n¹³ (1235520m⁷ + 25945920m⁶Δm + &c.)

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− h · Δm⁶/n¹⁵ (3603600m⁹ + &c.) + &c. (Δ⁶Q)

Ainsi des autres. Nota. — Ces expressions Δm², Δm³, &c., tiennent lieu de celles-ci (Δm)², (Δm)³, &c.” (Callet, pages 59 and 60.)

Other expressions for the differences of sin x are, if Δx = 2θ, as shown below,—

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Δ⁴ⁿ sin x = 2⁴ⁿ sin (x + 4nθ) sin⁴ⁿθ

Δ⁴ⁿ+1 sin x = 2⁴ⁿ+1 cos (x + 4n + 1 θ) sin⁴ⁿ⁺¹θ

Δ⁴ⁿ+2 sin x = − 2⁴ⁿ+2 sin (x + 4n + 2 θ) sin⁴ⁿ⁺²θ

Δ⁴ⁿ+3 sin x = − 2⁴ⁿ+3 cos (x + 4n + 3 θ) sin⁴ⁿ⁺³θ

(De Morgan, “Differential and Integral Calculus.”)

These four expressions may be combined into one expression as follows (Boole, “Finite Differences”):—

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Δ^m sin x = 2^m sin[x + m (π/2 + θ)] sin ^mθ

Now, Colenso (“Plane Trigonometry”) shows that when the tabular interval is small it is possible to use the expression