Background Trigonmetry (A)
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Complex Numbers Complex Analysis (BGND.1A) 3
Reference Angle (1) 1 st Quadrant Angle θ nd Quadrant Angle θ 90 180 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
Reference Angle () 3 rd Quadrant Angle θ 180 70 4 th Quadrant Angle θ 70 360 1 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
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
Symmetry http://en.wikipedia.org/wiki/derivative Complex Analysis (BGND.1A) 7
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
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
Shifts and periodicity http://en.wikipedia.org/wiki/derivative Complex Analysis (BGND.1A) 10
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
Angle Sum and Difference Identities (1) sin (60 + 30 ) = 1 1 y π α x x, y 1 sin (60 ) = cos(60 ) = 1 + 1 1 = 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
Angle Sum and Difference Identities () cos(30 + 60 ) = 0 1 y π α x x, y 1 sin (60 ) = cos(60 ) = 1 1 1 = 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 ) = 1 + 1 = Complex Analysis (BGND.1A) 13
Angle Sum and Difference Identities (3) tan(30 + 60 ) = + 1 y π α x x, y 1 tan(30 ) = 1 tan(60 ) = 1 + 1 1 = + 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 1 + 1 = 1 tan(60 ) = tan(30 ) = 1 Complex Analysis (BGND.1A) 14
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
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
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
References [1] http://en.wikipedia.org/ [] 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] www.chem.arizona.edu/~salzmanr/480a