Trigonometry (4 Trigonometric Identities 1//15
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Co-function Identities cos( π α = sin α y x, y sin( π α = cos α 1 π α x 1 tan( π α = cot α cot ( π α = tan α csc( π α = sec α sin = y cos( π α sec( π α = sec α cos = x sin( π α tan = y / x cot( π α Trigonometric Identities 3 1//15
ngle Sum and Difference Identities (1 sin (60 + 30 = 1 1 y π α x x, y 1 sin (60 = 3 cos(60 = 1 3 3 + 1 1 = 1 sin (30 = 1 cos(30 = 3 sin (60 30 = 1 sin(α + β = sin α cosβ + cos α sinβ sin(α β = sin α cosβ cos α sinβ sin (60 = 3 cos(60 = 1 sin (30 = 1 cos(30 = 3 3 3 1 1 = 1 Trigonometric Identities 4 1//15
ngle Sum and Difference Identities ( cos(30 + 60 = 0 1 y π α x x, y 1 sin (60 = 3 cos(60 = 1 3 1 1 3 = 0 sin (30 = 1 cos(30 = 3 cos(α + β = cosα cosβ sin α sin β cos(α β = cosα cosβ + sin α sin β cos(30 60 = 3 sin (60 = 3 cos(60 = 1 sin (30 = 1 cos(30 = 3 3 1 + 1 3 = 3 Trigonometric Identities 5 1//15
ngle Sum and Difference Identities (3 tan(30 + 60 = + 1 y π α x x, y 1 tan(30 = 1 3 tan(60 = 3 1 3 + 3 1 1 3 3 = + tan(60 = 3 tan(30 = 1 3 tan(α + β = tan(α β = tan(α + tan(β 1 tan(αtan(β tan(α tan(β 1 + tan(αtan(β tan(30 60 = 1 3 tan(30 = 1 3 tan(60 = 3 1 3 3 1 + 1 3 3 = 1 3 tan(60 = 3 tan(30 = 1 3 Trigonometric Identities 6 1//15
ngle Sum and Difference Identities (4 sin(α + β = sin α cosβ + cos α sinβ sin(α β = sin α cosβ cos α sinβ sin α sin β sin α sin β cos α cosβ cos α cosβ cos(α + β = cosα cosβ sin α sin β cos(α β = cosα cosβ + sin α sin β sin α sin β sin α sin β cos α cosβ cos α cosβ sin(α + β cos(α + β = sin α cosβ + cos α sin β cosα cosβ sin α sin β sin(α β cos(α β = sin α cosβ cos α sin β cosα cosβ + sin α sin β tan(α + β = tan(α + tan(β 1 tan(αtan(β tan(α β = tan(α tan(β 1 + tan(αtan(β Trigonometric Identities 7 1//15
Product to Sum (1 + sin(α + β = sin α cosβ + cos α sinβ + sin(α + β = sin α cosβ + cos α sinβ + sin(α β = sin α cosβ cos α sinβ sin(α β = sin α cosβ cos α sinβ sin(α + β + sin(α β = sin α cosβ sin(α + β sin(α β = cosα sin β sin α cosβ = 1 {sin(α + β + sin(α β} cos α sinβ = 1 {sin(α + β sin(α β} + cos(α + β = cosα cosβ sin α sin β + cos(α β = cosα cosβ + sin α sin β + cos(α + β = cosα cosβ sin α sin β cos(α β = cosα cosβ + sin α sin β cos(α + β + cos(α β = cosα cosβ cos(α + β + cos(α β = sin αsin β cos α cosβ = 1 {+ cos(α + β + cos(α β} sin α sin β = 1 { cos(α + β + cos(α β} Trigonometric Identities 8 1//15
Product to Sum ( sin(α ± β = sinα cosβ ± cos α sinβ sin(α + β = sin α cosβ + cos α sinβ sin(α β = sin α cosβ cos α sinβ cos(α ± β = cosα cosβ sin α sin β cos(α + β = cosα cosβ sin α sin β cos(α β = cosα cosβ + sin α sin β sin α cosβ = 1 {sin(α + β + sin(α β} sin α cosβ = 1 {+ sin(α + β + sin(α β} cos α sinβ = 1 {+ sin(α + β sin(α β} cos α cosβ = 1 {+ cos(α + β + cos(α β} cos α cosβ = 1 {+ cos(α + β + cos(α β} sin α sin β = 1 { cos(α + β + cos(α β} Trigonometric Identities 9 1//15
ngle sum and difference identities http://en.wikipedia.org/wiki/derivative Trigonometric Identities 10 1//15
Double ngle Formula http://en.wikipedia.org/wiki/derivative Trigonometric Identities 11 1//15
Triple-angle formulae http://en.wikipedia.org/wiki/derivative Trigonometric Identities 1 1//15
Half-angle formulae http://en.wikipedia.org/wiki/derivative Trigonometric Identities 13 1//15
Power-reduction formula http://en.wikipedia.org/wiki/derivative Trigonometric Identities 14 1//15
Product-to-sum http://en.wikipedia.org/wiki/derivative Trigonometric Identities 15 1//15
Sum-to-product http://en.wikipedia.org/wiki/derivative Trigonometric Identities 16 1//15
Euler's Formula e i θ = cos(θ + i sin(θ i( + e = cos( + + i sin ( + e i e i = (cos( + i sin( (cos( + i sin ( = [cos(cos( sin ( sin(] + i[cos( sin( + sin( cos(] sin( + = sin( cos( + cos( sin ( cos( + = cos( cos( sin( sin( Trigonometric Identities 17 1//15
Sin( angle sum and difference sin ( + sin ( sin( sin (cos( cos( sin ( cos( cos( sin ( sin ( sin ( sin (cos( cos( sin ( cos( cos( Trigonometric Identities 18 1//15
Cos( angle sum and difference cos( + sin ( sin( cos( cos( sin( sin ( cos( cos( cos( sin( sin ( cos( cos( + sin( sin( cos( cos( Trigonometric Identities 19 1//15
Product to Sum : sin cos sin ( + +sin ( sin( sin( sin ( sin( cos( cos( cos( cos( sin( + + sin( sin( cos( Trigonometric Identities 0 1//15
Product to Sum : cos sin sin( + sin( cos( sin( sin ( + sin( sin ( sin( sin ( sin( cos( cos( cos( cos( Trigonometric Identities 1 1//15
Product to Sum : cos cos cos( + +cos( sin ( sin( sin ( sin( cos( cos( cos( cos( cos( + + cos( cos( cos( Trigonometric Identities 1//15
Product to Sum : sin sin cos( + cos( sin( sin( cos( + + cos( +sin( sin( cos( + cos( sin ( sin( sin ( sin( cos( cos( cos( cos( Trigonometric Identities 3 1//15
Product to Sum sin ( + +sin ( sin( sin( sin ( sin( cos( cos( cos( cos( sin ( + sin( sin ( sin( sin ( sin( cos( cos( cos( cos( Trigonometric Identities 4 1//15
Product to Sum cos( + +cos( sin ( sin( sin ( sin( cos( cos( cos( cos( cos( + cos( sin ( sin( sin ( sin( cos( cos( cos( cos( Trigonometric Identities 5 1//15
Sum and Difference + = X = Y X +Y = ++ = X Y = + + = Trigonometric Identities 6 1//15
Product to Sum sin (X +sin (Y sin ( X +Y sin( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y sin (X sin(y sin ( X +Y sin( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y Trigonometric Identities 7 1//15
Product to Sum cos(x +cos(y sin ( X +Y sin( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y cos(x cos(y sin ( X +Y sin( X Y sin ( X +Y sin ( X Y cos( X +Y cos( X Y cos( X +Y cos( X Y X +Y X Y X +Y X Y Trigonometric Identities 8 1//15
Product-to-Sum & Sum-to-Product SUM sin( + + sin( sin( cos( sin( + sin( cos( sin( cos( + + cos( cos( cos( cos( + + cos( sin( sin( SUM PRODUCT PRODUCT sin(x + sin(y sin( X +Y cos( X Y sin(x sin(y cos( x+y X Y sin( cos(x + cos(y cos( X +Y cos( X Y cos( X + cos(y sin( X +Y sin( X Y Trigonometric Identities 9 1//15
References [1] http://en.wikipedia.org/ [] http://planetmath.org/ [3] litzer, R. lgebra & Trigonometry. 3rd ed, Prentice Hall [4] Smith, R. T., Minton, R.. Calculus: Concepts & Connections, Mc Graw Hill [5] 홍성대, 기본 / 실력수학의정석, 성지출판 1//15