Trigonometry (4A) Trigonometric Identities. Young Won Lim 1/2/15
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1 Trigonometry (4 Trigonometric Identities 1//15
2 Copyright (c Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no ack-cover Texts. copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to This document was produced by using OpenOffice and Octave. 1//15
3 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
4 ngle Sum and Difference Identities (1 sin ( = 1 1 y π α x x, y 1 sin (60 = 3 cos(60 = = 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 = = 1 Trigonometric Identities 4 1//15
5 ngle Sum and Difference Identities ( cos( = 0 1 y π α x x, y 1 sin (60 = 3 cos(60 = = 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 Trigonometric Identities 5 1//15
6 ngle Sum and Difference Identities (3 tan( = + 1 y π α x x, y 1 tan(30 = 1 3 tan(60 = = + 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 = = 1 3 tan(60 = 3 tan(30 = 1 3 Trigonometric Identities 6 1//15
7 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
8 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
9 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
10 ngle sum and difference identities Trigonometric Identities 10 1//15
11 Double ngle Formula Trigonometric Identities 11 1//15
12 Triple-angle formulae Trigonometric Identities 1 1//15
13 Half-angle formulae Trigonometric Identities 13 1//15
14 Power-reduction formula Trigonometric Identities 14 1//15
15 Product-to-sum Trigonometric Identities 15 1//15
16 Sum-to-product Trigonometric Identities 16 1//15
17 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
18 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
19 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
20 Product to Sum : sin cos sin ( + +sin ( sin( sin( sin ( sin( cos( cos( cos( cos( sin( + + sin( sin( cos( Trigonometric Identities 0 1//15
21 Product to Sum : cos sin sin( + sin( cos( sin( sin ( + sin( sin ( sin( sin ( sin( cos( cos( cos( cos( Trigonometric Identities 1 1//15
22 Product to Sum : cos cos cos( + +cos( sin ( sin( sin ( sin( cos( cos( cos( cos( cos( + + cos( cos( cos( Trigonometric Identities 1//15
23 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
24 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
25 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
26 Sum and Difference + = X = Y X +Y = ++ = X Y = + + = Trigonometric Identities 6 1//15
27 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
28 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
29 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
30 References [1] [] [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
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