交流電與正弦波 a b N c S d i V t 8 4 8

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3 交流電與正弦波 a b N c S d i V t 8 4 8

4 Joseph Fourier P xy P rop x y sin y r cos x r y P x,y tan y x x0 cot x y y0 sec r x x0 csc r y y0 y x r O θ x y P sin y ycos x x P xycos sin P y sin P cossin cos sin θ 0 x

5 y=sinx 4 ysinx 4 4 P P xy sincos P x,y θ 0 4 r P P xy sincos

7 7 Geogebra 4~57 x 8 繪圖區 8 9 x 9 x

8 8 ysinx 0ysin xenter ysinx 0 y=sin x y=sinx sin xsinx f px f x p ysinx

9 9 點選的物件 4 4

10 0 - Geogebra y= a sinx Geogebra ysinx ysinx ysinx 4ysinx 4 ysinx ysinx ysinx ysinx

11 Geogebra yasinx 5 yasinxya*sin x 繪圖區 ya*sin x 5 a yasinx 49~5 yasinx yasinx yasinx yasinx 4 yasinx a

12 ia sina í θ

13 Geogebra ysin bx Geogebra ysinx ysinx ysin4x 4ysin x 4 ysinx ysinx ysin4x ysin x

14 4 Geogebra ysinbx b8 b 8 ysinbx ysinbx 5~54 ysinbx ysinbx ysinbx ysin bx 4 ysinbx b

15 5 Hz / t 00t0 v t00sin 00t0 T / f Hz i ti m sin f t 0 t I m 0 0

16 6 4 Geogebra ysinx+c Geogebra ysinx ysin x ysin x 4ysin x 4 ysinx ysin x ysin x ysin x

17 7 Geogebra sin xc c5 c 5 ysin xc ysin xc 54~55 ysin xc ysin xc ysin xc ysinx 4 ysin xcc

18 8 i t I m sin t+ 0 I m sin f t 0 0 I m f Hz / f / t 0 f t 0 0 v t v t Av t v t 05 Bv t v t 05 Cv t v t 05 Dv t v t v V 4560 v t v t 0.0s t s v t60sin 77t0 i t40sin 77t0 Av i 0 Bv i 0 Cv i 40 Dv i 40

20 0 6 Geogebra yasinbx+d Geogebra ysinx 55~56 Geogebra ysinx4 ysinx ysinx4 ysinx ysinx4 ysinx ysinx4

21 Geogebra ysinx Geogebra ysinx ysinx ysinx ysinx ysinx ysinx ysinx yasin bx+c+d abcd

22 想想看前面曾經提過在電學中交流電的電壓電流等常可以寫成下面的式子 x t A sin ωt+θ0 A sin πf t θ0 t 0 其中 A 振幅 f 頻率 Hz 週期運動的次數 / 秒 t 時間秒 ω πf 角速度弧度 / 秒 θ0 初相角線圈初始的位置 πf t θ0 相位角下圖為某電壓的波形請問以下何者近似此電壓瞬間值 v t 的關係式 A v t 00sin 77t B v t 70.7sin 77t C v t 00sin 4t Dv t 00sin 77t 60 伏特

23 v t t v t00sin 0t 0 A 60 B 0 C 60 D 0 v t t v t00sin 0t 0 A 0 B 5 C 60 D 0

24 4 yf xsin x Ayf xsin x Byf xsin x Cyf xsin x Dyf xsin x y 0 x v i v i A0 B45 C5 D60 i V t v t0sin t i t Ai t7sin t45 Bi t7sin t45 Ci t0sin t45 Di t7sin t60 5 x ta sin t 0 A sin f t 0 t 0 A f Hz / t f / 0 f t 0

25 5 v t Av t0sin 50t45 Bv t0sin 50t45 Cv t0sin 00t45 Dv t0sin 00t v V s v t v t t s 6 e te m sin t0i ti m cos t0 e i Ae i 60 Be i 50 Ci e 0 Di e 60 7 v t i t v t0 sin 4 t0 i t0 sin 4 t80 A90 B0 C90 D0 8 i t t i ti m sin t 6 0 At Bt Ct 4 Dt 9 At 4 Bt Ct 4 Dt 0 4 e t Ae te m sint e Be te m sint Em π Ce te m sin t90 π De te m sin t90 Em 4 t q e00sin t60 i0sin t0 A 40 B 40 C 80 D 80

26 6 w v t00sin 77t0 i t5sin 77t60 Av i 0 Bv i 0 Cv i 90 Dv i 90 e t x t0sin 0t 60 Ax t0cos 0t60 Bx t 0cos 0t 50 Cx t0cos 0t0 Dx t0cos 0t60 r v t t v t0 sin 77t 0A 60 B 50 C 4 D00 Hz t i t t i t00sin 77t0 i ta 5 B 50 C 60 D 75 / yf xsin xsin x y 0 x C B 4A 5B 6C 7D 8A 9D 0C qb wb v i 0600 ec ra tc

27 V 0 00 Yahoo 005 Yahoo Alternating Current AC 0 t θt θ V m v t 0 θ 4 t θ t T V m T

28 8 P W Q V P W QV t t WQV I I Q t WQV Q t VtI V t VIR W IVtIIRtI R t mAh WPt P

29 9 y y 4, y 4 y4,4 x4 x O 4 x 4 O x y 4, y y 4 x x4 x O 4 x O x y y, xy0 x x O x O x

31 58 4 Geogebra ysinx x 664 x 4 ysinx ysinx Geogebra 5 Geogebra ysinx x 4 x

32 45 V av I av x x 0 0 y 0 0 x x x

33 Geogebra 664 ysinx ysinx x0 x v tv m sin t V m t 4 A v t V m O A v tv m sin t t 4 x A V av i ti m sin t I av

34 4 v t00sin t a b 00 v 0 t 00 v a 0 t b T R R effective value V eff I eff

35 5 R R I eff I DC T C T C 5 8 WI Rt W I m sin t Rt I m sin ti m I I m sin t cossin sin cos I I m cost I m cost I m

36 6 6 Geogebra Geogebra 0 I I m cost I m I eff v tv m sin t V eff Geogebra I I m sin t I 7 ysin x x0 x Geogebra.57 ysin x

37 7 7 Geogebra ysin x I I m sin t I m I eff I m 4 v t00sin 60t v t t t

38 8 A B C D AC0V A B C D 5V 0.5W A5 B0 C.5 D50 4 v t00 sin 0t A90 B00 C0 D0 5 0V / 60Hz v t Av t0sin 60t Cv t56sin 60t Bv t0sin 77t Dv t56sin 77t 00 B V m V av C V m V m A 4B 5Dv tv eff sint0 sin 60t56sin77t V

39 9 Geogebra Geogebra Markus Hohenwarter Geogebra Geometry Algebra Geogebra Geogebra Geogebra JAVA JAVA Geogebra JAVA 40 Geogebra Enter

40 40 功能表列工具列代數區繪圖區指令區 Geogebra yx Enter a yx a a 輸入指令後, 代數區顯示其代數式, 繪圖區出現其圖形在指令列輸入 y=x+ 後按下 Enter 鍵 Geogebra Step Step yx Step Enter Step 4 ayx

41 4 Geogebra - Geogebra Geogebra C,5 C,5 yx x5 yx ^ x5 x x ^ yx yabs x abs ysinx ysin x x a,4 a,4 ^ Shift 6 4 Enter C a yx x5 c yx f x ysinx g x Geogebra

42 Geogebra Geogebra B, y x yxx ycosx

43 x sinx y, 0,0 / / x, 7 ysinx

44 44 Geogebra x 8 繪圖區 8 x 9 9 x

45 45 ysinx ysin x 0 0 ysin x Enter ysinx ysinx

46 46 sin xsinx f pxf x p ysinx ysinx R ysinx y y ysinx

47 47 y=cosx x cosx y 0,,0 0 / /,0 x, ycosx

48 48 Geogebra ycosx ycos x Enter ycosx ycosx ycosx R ycosx y y ycosx ycosx ysinx

49 49 Geogebra yf x yaf bxcd. yasinx ysinx Geogebra ya sinx ysinx ysinx 4 繪圖區 4 a

50 50 5 數值滑桿 5 yasinx y a*sin x Enter 6 ya*sin x 6 yasinx

51 5 7 點選的物件

53 5 yasinx ysinx a yasinx yasinx ysinx a 4 y4sinx ysinx ysinx 0 π/ 0 π/ π π/ π 5π/ π 7π/ 4π 9π/ ysinx 4 y=asinx. ysin bx ysin x ysinx b b 5

54 54 在繪圖區按一下 b ysinbx ysin b*x Enter ysin b*x y=sinbx ysinbx ysinx b ysinbx ysinbx ysinx b

55 55. ysinx ysin x+c ysinx c c 4 在繪圖區按一下 4 c ysin xcysin xc Enter 56 5

56 56 ysin xc 5 ysin x+c ysin xc ysinx c ysin xc c0ysin xc ysinx c c0ysin xc ysinx c 4. ysinx ysinx+d ysinx d d 57 6

57 57 在繪圖區按一下 6 d ysinxd ysinxd Enter 7 ysinxd 7 ysinxd ysinxd ysinx d ysinxd d0ysinxd ysinx d d0 ysinxd ysinx d

58 58 Geogebra Geogebra ysinx x Step Step x 8 Step ysin x Enter x

59 59 9Geogebra ysinx 9 Geogebra ysinx f xsin x Geogebra ysinx f x Geogebra x ab75 5 ab 5

60 60 y 5 a7 4 b x 0 f x a x b f x x xa xb f x a b b f x a b a f xdx f x a b

61 6 f x a x b x f x a x b x x f x y a b x x Geogebra f x x x4 f x y 4 f x x x4 0 4 f x x4 f x 0 4 x

62 6 Geogebra Geogebra y0.5x Enter ay0.5x Integral [ a, 0, 4 ] Enter x x0 y x4 b Integral [ a, 0, 4 ] Intergral a y0.5x 0 x0 4 x4 Geogebra f x x4

63 6 Geogebra ysinx x Geogebra ysinx x Integral [ f, 0, pi ] Enter ysinx x a 4 4 ysinx x0 x Integral [ f, 0, pi ] Intergral f f xsinx 0 x0 pi x Geogebra ysinx

64 64 Geogebra ysinx x x Step ysinx x Step Integral[f, pi, pi] Enter Step ysinx x 5 Step 4 a 5 ysinx x x Geogebra ycosx x0 x Geogebra

65 65 Geogebra x ysinx 0 x ysinx x yx 0 x yx x yx x Integral fgh

67 素養導向數學教材 / 曾世杰主編 -- 初版 -- 新北市三峽區 : 國家教育研究院. 數學教育. 中小學教育. 教材與教法素養導向技術型高級中學數學教材 : 交流電中的數學主編者 : 單維彰作者 : 馬雅筠陳吳煜 ( 依姓氏筆畫順序排列 ) 發行人 : 柯華葳出版者 : 國家教育研究院編審者 : 十二年國民基本教育數學素養教材研發編輯小組召集人 : 曾世杰副召集人 : 單維彰鄭章華編輯小組 : 古欣怡朱安強林美曲林信安馬雅筠陳吳煜陳淑娟曾明德曾俊雄鄧家駿 ( 依姓氏筆畫順序排列 ) 版次 : 初版電子全文可至國家教育研究院網站 http: // 免費取用本書經雙向匿名審查通過 ( 歡迎使用, 請註明出處 )

Geogebra A B C S Geogebra 1

Geogebra A B C S Geogebra 1 Yahoo 110 110 Geogebra 0.707 Geogebra Geogebra 3 P P5 300 150 100 Hipparchus of Nicaea 1 AB A AM sin = AM θ M 1 0 θ 1 Euler 1 B 1 sinus (cosine) complemental sine : (John Napier)

交流電 與正弦波 a b N c S d i V t 8 4 8

交流電與正弦波 a b N c S d i V t 8 4 8