Background Trigonmetry (2A) Young Won Lim 5/5/15

Transcription

1 Background Trigonmetry (A)

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 Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Complex Numbers Complex Analysis (BGND.1A) 3

4 Reference Angle (1) 1 st Quadrant Angle θ nd Quadrant Angle θ y x, y x, y y 1 x 1 1 x 1 Reference Angle α = 180 sin = y cos = x tan = y / x sin θ = + sin α = (+ y) cos θ = cos α = ( x) tan θ = tan α = (+ y)/( x) Complex Analysis (BGND.1A) 4

5 Reference Angle () 3 rd Quadrant Angle θ th Quadrant Angle θ x 1 1 x 1 x, y y y x, y Reference Angle α = 180 sin θ = sin α = ( y) cos θ = cos α = ( x) tan θ = + tan α = ( y)/( x) Reference Angle α = 360 sin θ = sin α = ( y) cos θ = + cos α = (+ x) tan θ = tan α = ( y)/(+ x) Complex Analysis (BGND.1A) 5

6 Reference Angle (3) A Quadrant Angle θ Reference Angle α = sin = sin cos = cos tan = tan = sin = sin cos = cos tan = tan only sin + x, y All + x, y sin = sin cos = cos tan = tan = sin = sin cos = cos tan = tan = x, y only tan + only cos + x, y Complex Analysis (BGND.1A) 6

7 Symmetry Complex Analysis (BGND.1A) 7

8 Shift by +π/ cos(α + π ) = sin α x, y sin(α + π ) = +cos α ( y, +x) α + π tan(α + π ) = cot α cot(α + π ) = tan α csc (α + π ) = +sec α sec(α + π ) = csc α Complex Analysis (BGND.1A) 8

9 Shift by -π/ cos(α π ) = +sin α x, y sin(α π ) = cos α α π tan(α π ) = cot α cot(α π ) = tan α (+ y, x) csc (α π ) = sec α sec(α π ) = +csc α Complex Analysis (BGND.1A) 9

10 Shifts and periodicity Complex Analysis (BGND.1A) 10

11 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) = csc α cos = x sin( π α) tan = y / x cot( π α) Complex Analysis (BGND.1A) 11

12 Angle Sum and Difference Identities (1) sin ( ) = 1 1 y π α x x, y 1 sin (60 ) = cos(60 ) = = 1 sin (30 ) = 1 cos(30 ) = sin (60 30 ) = 1 sin(α + β) = sin α cosβ + cos α sinβ sin(α β) = sin α cosβ cos α sinβ sin (60 ) = cos(60 ) = 1 sin (30 ) = 1 cos(30 ) = 1 1 = 1 Complex Analysis (BGND.1A) 1

13 Angle Sum and Difference Identities () cos( ) = 0 1 y π α x x, y 1 sin (60 ) = cos(60 ) = = 0 sin (30 ) = 1 cos(30 ) = cos(α + β) = cosα cosβ sin α sin β cos(α β) = cosα cosβ + sin α sin β cos(30 60 ) = sin (60 ) = cos(60 ) = 1 sin (30 ) = 1 cos(30 ) = = Complex Analysis (BGND.1A) 13

14 Angle Sum and Difference Identities (3) tan( ) = + 1 y π α x x, y 1 tan(30 ) = 1 tan(60 ) = = + tan(60 ) = tan(30 ) = 1 tan(α + β) = tan(α β) = tan(α) + tan(β) 1 tan(α)tan(β) tan(α) tan(β) 1 + tan(α)tan(β) tan(30 60 ) = 1 tan(30 ) = 1 tan(60 ) = = 1 tan(60 ) = tan(30 ) = 1 Complex Analysis (BGND.1A) 14

15 Angle 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(α)tan(β) tan(α β) = tan(α) tan(β) 1 + tan(α)tan(β) Complex Analysis (BGND.1A) 15

16 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 β) = sin α cosβ + sin(α + β) = sin α cosβ + cos α sinβ sin(α β) = sin α cosβ cos α sinβ 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)} Complex Analysis (BGND.1A) 16

17 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)} Complex Analysis (BGND.1A) 17

18 References [1] [] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [5]