Trigonometr (3) Qudrnt ngle Trigonometr Negtive ngle Trigonometr Referene ngle Trigonometr Sinusoidl Wves
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Tringle Trigonometr Right Tringle Olique Tringle ll ute ngles One Otuse ngle 0 90 0 90 90 180 sin = / os = / tn = / sin =? os =? tn =? sin =? os =? tn =? Trigonometr (3) 3
Olique Tringles Trigonometr ll ute ngles sin = sin180 = sin os = os180 = os tn = tn 180 = tn 0 90, 0 180 One Otuse ngle The Lw of Sines sin = sin = sin The Lw of osines 2 2 = 2 2 2 os = 2 2 2 os 2 = 2 2 2 os Trigonometr (3) 4
One Otuse ngle (1) The Lw of Sines sin = sin = sin α h h α α sin = sin = h = h sinα = h sin = sinα Trigonometr (3) 5
One Otuse ngle (2) The Lw of osines 2 = 2 2 2 os α os = ( 2 + 2 2 ) / 2 2 = h 2 + 2 os 2 α 2 = h 2 + (osα + ) 2 h α osα h 2 (osα + ) 2 h α (osα + ) ( 2 + 2 2 ) = h 2 + 2 os 2 α + 2 h 2 ( osα + ) 2 os = 2 os α / 2 os = os α Trigonometr (3) 6
Trigonometr in the 2 nd Qudrnt ngles (1) (-x, ) (+x, + ) -x β = 180 α x sin = sin180 = sin os = os180 = os tn = tn 180 = tn sin = / os = x / tn = / x Trigonometr (3) 7
Trigonometr in the 2 nd Qudrnt ngles (2) (-x, +) + (-x, ) -x β = 180 α -x sin = sin180 = sin os = os180 = os tn = tn 180 = tn sin β = (+ ) / osβ = ( x) / tnβ = (+ )/( x) Trigonometr (3) 8
Trigonometr in the 2 nd Qudrnt ngles (3) Isoseles Tringle (-x, ) (-x, ) -x -x r r +r r = x 2 2 sin = / os = x / tn = / x sinβ = (+ ) / r osβ = ( x) / r tnβ = (+ ) / ( x) Trigonometr (3) 9
Trigonometr in Qudrnt ngles (4) Unit irle (-x, ) ( x, ) r -x r +r 1 x 1 r = x 2 2 sin = / r os = x / r tn = / x 1 = x 2 + 2 sin β = +sin α = (+ ) osβ = sin β = ( x) tnβ = tn α = (+ )/( x) Trigonometr (3) 10
Negtive ngle Trigonometr (1) 1 st Qudrnt ngle 4 th Qudrnt ngle (+x, + ) 1 x 1 1 x 1 0 < α < 90 90 < α < 0 (+x, ) sin α = (+ ) os α = (+ x) tn α = (+ )/(+ x) sin( α) = sin α = ( ) os( α) = + os α = (+x) tn( α) = tn α = ( )/(+x) Trigonometr (3) 11
Negtive ngle Trigonometr (2) 2 nd Qudrnt ngle 3 rd Qudrnt ngle ( x, + ) 1 x 1 1 x 1 ( x, ) 90 < β < 180 180 < β < 90 sin β = + sin α = (+ ) osβ = os α = ( x) tnβ = tn α = (+ )/( x) sin( β) = sin α = ( ) os( β) = os α = ( x) tn( β) = + tn α = ( )/( x) Trigonometr (3) 12
Referene ngle (1) 1 st Qudrnt ngle θ 2 nd Qudrnt ngle θ 90 180 x, x, 1 x 1 1 x 1 Referene ngle α = 180 sin = os = x tn = / x sin θ = + sin α = (+ ) os θ = os α = ( x) tn θ = tn α = (+ )/( x) Trigonometr (3) 13
Referene ngle (2) 3 rd Qudrnt ngle θ 180 270 4 th Qudrnt ngle θ 270 360 1 x 1 1 x 1 x, x, Referene ngle α = 180 sin θ = sin α = ( ) os θ = os α = ( x) tn θ = + tn α = ( )/( x) Referene ngle α = 360 sin θ = sin α = ( ) os θ = + os α = (+ x) tn θ = tn α = ( )/(+ x) Trigonometr (3) 14
Referene ngle (3) Qudrnt ngle θ Referene ngle α = sin = sin os = os tn = tn = sin = sin os = os tn = tn onl sin + x, ll + x, sin = sin os = os tn = tn = sin = sin os = os tn = tn = 2 x, onl tn + onl os + x, Trigonometr (3) 15
Mking Helix Trnsprent OHP Film Trigonometr (3) 16
Helix nd Viewpoints Top View x x z z Side View x z Front View Trigonometr (3) 17
Sine Wve z 0 2 3 2 x Side View Sine Wve z Front View Trigonometr (3) 18
osine Wve z 0 2 3 2 x Side View osine Wve z Top View Trigonometr (3) 19
Smmetr in Sinusoid Trigonometr (3) 20
Sine nd osine Wves 1 2 0 1 2 2 3 2 5 2 3 Sine Wve osine Wve Trigonometr (3) 21
Sine Wve Smmetr Sine Wve 1 2 1 2 3 2 5 2 0 2 3 Trigonometr (3) 22
osine Wve Smmetr osine Wve 1 2 1 2 3 2 5 2 0 2 3 Trigonometr (3) 23
Referenes [1] http://en.wikipedi.org/ [2] http://plnetmth.org/ [3] litzer, R. lger & Trigonometr. 3rd ed, Prentie Hll [4] Smith, R. T., Minton, R.. lulus: onepts & onnetions, M Grw Hill [5] 홍성대, 기본 / 실력수학의정석, 성지출판