Development of trigonometry Medieval times Serge G. Kruk/Laszl´ o´ Liptak´ Oakland University Development of trigonometry Medieval times – p.1/27 Indian Trigonometry • Work based on Hipparchus, not Ptolemy • Tables of “sines” (half chords of twice the angle) • Still depends on radius 3◦ • Smallest angle in table is 3 4 . Why? R sine( α ) α Development of trigonometry Medieval times – p.2/27 Indian Trigonometry 3◦ • Increment in table is h = 3 4 • The “first” sine is 3◦ 3◦ s = sine 3 = 3438 sin 3 = 225 1 4 · 4 1◦ • Other sines s2 = sine(72 ), . Sine differences D = s s • 1 2 − 1 • Second-order approximation techniques sine(xi + θ) = θ θ2 sine(x ) + (D +D ) (D D ) i 2h i i+1 − 2h2 i − i+1 Development of trigonometry Medieval times – p.3/27 Etymology of “Sine” How words get invented! • Sanskrit jya-ardha (Half-chord) • Abbreviated as jya or jiva • Translated phonetically into arabic jiba • Written (without vowels) as jb • Mis-interpreted later as jaib (bosom or breast) • Translated into latin as sinus (think sinuous) • Into English as sine Development of trigonometry Medieval times – p.4/27 Arabic Trigonometry • Based on Ptolemy • Used both crd and sine, eventually only sine • The “sine of the complement” (clearly cosine) • No negative numbers; only for arcs up to 90◦ • For larger arcs, the “versine”: (According to Katz) versine α = R + R sine(α 90◦) − Development of trigonometry Medieval times – p.5/27 Other trig “functions” Al-B¯ırun¯ ¯ı: Exhaustive Treatise on Shadows • The shadow of a gnomon (cotangent) • The hypothenuse of the shadow (cosecant) • The reverse shadow (tangent) • The hypothenuse of the reverse shadow (secant) sun sun reverse shadow shadow Development of trigonometry Medieval times – p.6/27 Height and distance via sine Al-Qab¯ıs¯ı: • Given α1; α2; d • (angles via astrolabe) • find x; y using only sines A y E α1 α2 C d D B x Development of trigonometry Medieval times – p.7/27 Height and distance via sine d sin α y = 2 sin(90◦ α1) sin α2 sin(90◦ α2) − − − sin α1 y sin(90◦ α ) x = − 1 sin α1 Development of trigonometry Medieval times – p.8/27 Arabic Trigonometry • Many trig identities • “Formal” development • Both plane and spherical trig Development of trigonometry Medieval times – p.9/27 Regiomontanus 1436–1476 • Real name Johannes Müller • Translated Ptolemy • Re-did Ptolemy from basic axioms (à la Euclid) • Plane and spherical trig • Many results based on arabic texts Development of trigonometry Medieval times – p.10/27 Later works • Tables of all trig functions n • Based on larger and larger radii (10 ) • . for accuracy • . no good notation for decimals Development of trigonometry Medieval times – p.11/27 Names for trig functions • Regiomontanus 1436–1476: On Triangles of Every Kind • Sinus complementi • Joachim Rheticus 1514–1574 • Trig functions from triangles, no circles! • Thomas Fincke 1561–1656 • Invented words “tangent” and “secant” • Bartholomeo Pitiscus 1561–1613 • Invents the word “Trigonometry” Everything still depends on a particular radius Development of trigonometry Medieval times – p.12/27 Pitiscus's front page Development of trigonometry Medieval times – p.13/27 From • A Gentle History for Teachers and Others • by Berlinghoff and Gouvêa Development of trigonometry Medieval times – p.14/27 Versine • Isaac Newton's Principia 1713 • Charles Hutton A course in Mathematics 1801 • Sine: The line drawn from one extremity of an arc to the diameter, perpendicular to the diameter. • Versed sine: Part of the diameter intercepted between an arc and its sine. R sine( α ) α versine( α ) Development of trigonometry Medieval times – p.15/27 Why tangent? Why secant? Similar triangles (Circle of radius R = 1) a c tan α = = and c is tangent b R R d sec α = = and d is secant b R d R c a α b R Development of trigonometry Medieval times – p.16/27 Napier 1614 • Our notation for decimals • Logs to simplify trig calculations • (Replacing multiplication of sines by addition) Development of trigonometry Medieval times – p.17/27 Leibniz (1693) and sine • Consider a unit circle • Let x = sin y dt dy dx 1 y x Development of trigonometry Medieval times – p.18/27 x = sin y • By similar triangles (because dy is tangent) dt x = dx p1 x2 − • By right angle triangle dx2 + dt2 = dy2 • Isolate dt from second, sub into first, square (1 x2)(dy2 dx2) = x2dx2 − − dy2 = x2dy2 + dx2 Development of trigonometry Medieval times – p.19/27 dy2 = x2dy2 + dx2 • Differential, assume dy constant 0 = d(x2dy2 + dx2) = 2 d(x dy) x dy + 2 d(dx) dx 2 2 = 2xdy(dx dy + x d y) +2 dx dx = 2 dx(d2x + x dy2) • Therefore d2x = x dy2 − Development of trigonometry Medieval times – p.20/27 2 d x = x dy2 − • Since sin is an odd function • and since sin 0 = 0 • Liebniz assumes a power series x = y + cy3 + ey5 + fy7 + : : : • Differentiate twice d2x = 3 2cy + 5 4ey3 + 7 6fy5 + : : : dy2 · · · • Equate term by term both series Development of trigonometry Medieval times – p.21/27 x = sin y • Finally 1 1 x = sin y = y y3 + y5 : : : − 3! 5! − • An expression for sin free of the circle Development of trigonometry Medieval times – p.22/27 Euler 1730 • Power expansion of exponentials and log • Power expansion of sine and cosines • States matter-of-factly a unit circle • And relations between all of them • Trig function really as “functions” • (For Euler, that means analytic) Development of trigonometry Medieval times – p.23/27 Euler: Unification iθ iθ e = cos θ + i sin θ e− = cos θ i sin θ − iθ iθ iθ iθ e e− e + e− sin θ = − cos θ = 2i 2 θ θ θ θ e e− e + e− sinh θ = − cosh θ = 2 2 Name due to Lambert : “Hyperbolic trig functions” Development of trigonometry Medieval times – p.24/27 Why Hyperbolic ? cosh(a) x^2−y^2 = 1 sinh(a) 1 Development of trigonometry Medieval times – p.25/27 Hyperbolic trig sinh θ tanh θ = cosh θ cosh θ coth θ = sinh θ 1 sec hθ = cosh θ 1 csc hθ = sinh θ Verify cosh2θ sinh2θ = 1 − Development of trigonometry Medieval times – p.26/27 Trig: the big picture • Original concept: chords (depends on radius) • Use: Solving triangles (plane and spherical) • Middle-age “sine”: half-chord of double angle • Liebniz and Euler: Unit circle • Euler: trig as proper functions • One Family: exponentials, logs, trig, hyperbolics Discussion: What is a definition for a mathematician? Development of trigonometry Medieval times – p.27/27.