Trigonometry (3A) Quadrant Angle Trigonometry Negative Angle Trigonometry Reference Angle Trigonometry Sinusoidal Waves. Young Won Lim 12/30/14

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1 Trigonometr (3) Qudrnt ngle Trigonometr Negtive ngle Trigonometr Referene ngle Trigonometr Sinusoidl Wves

2 opright () Young W. Lim. Permission is grnted to op, distriute nd/or modif this doument under the terms of the GNU Free Doumenttion Liense, Version 1.2 or n lter version pulished the Free Softwre Foundtion; with no Invrint Setions, no Front-over Texts, nd no k-over Texts. op of the liense is inluded in the setion entitled "GNU Free Doumenttion Liense". Plese send orretions (or suggestions) to oungwlim@hotmil.om. This doument ws produed using OpenOffie nd Otve.

3 Tringle Trigonometr Right Tringle Olique Tringle ll ute ngles One Otuse ngle sin = / os = / tn = / sin =? os =? tn =? sin =? os =? tn =? Trigonometr (3) 3

4 Olique Tringles Trigonometr ll ute ngles sin = sin180 = sin os = os180 = os tn = tn 180 = tn 0 90, One Otuse ngle The Lw of Sines sin = sin = sin The Lw of osines 2 2 = os = os 2 = os Trigonometr (3) 4

5 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

6 One Otuse ngle (2) The Lw of osines 2 = os α os = ( ) / 2 2 = h os 2 α 2 = h 2 + (osα + ) 2 h α osα h 2 (osα + ) 2 h α (osα + ) ( ) = h os 2 α + 2 h 2 ( osα + ) 2 os = 2 os α / 2 os = os α Trigonometr (3) 6

7 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

8 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

9 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

10 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 sin β = +sin α = (+ ) osβ = sin β = ( x) tnβ = tn α = (+ )/( x) Trigonometr (3) 10

11 Negtive ngle Trigonometr (1) 1 st Qudrnt ngle 4 th Qudrnt ngle (+x, + ) 1 x 1 1 x 1 0 < α < < 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

12 Negtive ngle Trigonometr (2) 2 nd Qudrnt ngle 3 rd Qudrnt ngle ( x, + ) 1 x 1 1 x 1 ( x, ) 90 < β < < 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

13 Referene ngle (1) 1 st Qudrnt ngle θ 2 nd Qudrnt ngle θ 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

14 Referene ngle (2) 3 rd Qudrnt ngle θ th Qudrnt ngle θ 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

15 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

16 Mking Helix Trnsprent OHP Film Trigonometr (3) 16

17 Helix nd Viewpoints Top View x x z z Side View x z Front View Trigonometr (3) 17

18 Sine Wve z x Side View Sine Wve z Front View Trigonometr (3) 18

19 osine Wve z x Side View osine Wve z Top View Trigonometr (3) 19

20 Smmetr in Sinusoid Trigonometr (3) 20

21 Sine nd osine Wves Sine Wve osine Wve Trigonometr (3) 21

22 Sine Wve Smmetr Sine Wve Trigonometr (3) 22

23 osine Wve Smmetr osine Wve Trigonometr (3) 23

24 Referenes [1] [2] [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