Sine Cosine functions Beijing 170408 jmlecolealsacienne org 1
Sine & Cosine functions Beijing - 17/04/08 jml@ecole-alsacienne. org 1
Construction of the Sine function Beijing - 17/04/08 jml@ecole-alsacienne. org 2
Properties of the sine function : 1. Period : T = 2π Sin(x + 2π) = Sin(x + 4π) = … = sin(x + k. 2π) REMEMBER ! in PHYSICS the period can be either : • a period of time (like in a pendulum movement) • a period of space (like in a sine wave) Beijing - 17/04/08 jml@ecole-alsacienne. org 3
Properties of the sine function : 2. Maximum : y = sin (π/2) = 1 Sin(π/2 + 2π) = Sin(π/2 + 4π) = … = sin(π/2 + k. 2π) = 1 Beijing - 17/04/08 jml@ecole-alsacienne. org 4
Properties of the sine function : 3. Minimum : y = sin (3π/2) = -1 Sin(3π/2 + 2π) = Sin(3π/2 + 4π) = … = sin(3π/2 + k. 2π) = -1 Beijing - 17/04/08 jml@ecole-alsacienne. org 5
Properties of the sine function : 4. Symetry with respect to 0 Sin(-x) = - Sin(x) Sin(-x – k. 2π) = - Sin(x + k. 2π) Note : any intersection point with Ox is a center of Symetry. Beijing - 17/04/08 jml@ecole-alsacienne. org 6
Properties of the sine function : 5. Unchanged by any translation of k. 2π along the Ox axis Sin(x + k 2π) = Sin(x) That is to justify the construction of the curve by copying any part of length = 2π as many times a we can. Beijing - 17/04/08 jml@ecole-alsacienne. org 7
Properties of the sine function : 6. For the same variation ∆x the variation ∆y is much smaller around the maximum and the minimum. To see how it moves press this kee : (Now you understand why the days change less quickly in december and june than in march or october…) Beijing - 17/04/08 jml@ecole-alsacienne. org 8
Properties of the sine function : 8. For values of x close to 0, sin x ≈ x Beijing - 17/04/08 jml@ecole-alsacienne. org 9
Transfert from Sine to Cosine Cos x = Sin(x + π/2) Sin x = Cos(π/2 - x) Beijing - 17/04/08 jml@ecole-alsacienne. org 10
General Sine functions f(x) = Asin(ax + b) A=amplitude a =2π/T , T = period =2π /a. PROOF ? b =constant phase (ax +b) = phase Beijing - 17/04/08 jml@ecole-alsacienne. org 11
Combinations of sine functions y 1=2. sin(2π/3)x. . . ………. . Period : T 1 = 3 y 2=3. sin(πx) ……………. Period : T 2 = 2 y 3 =y 1 + y 2 = 2. sin(2πx/3) + 3 sin(πx) Prove that the Period T 3 = LCM (T 1 ; T 2) = 2 x 3 = 6 Beijing - 17/04/08 jml@ecole-alsacienne. org 12
Fourier’s theorem 1. The sum of periodic functions is also a periodic function. 2. Any periodical function can be written as the sum of a series of sine functions. Even a square signal … Beijing - 17/04/08 jml@ecole-alsacienne. org 13
Other kind of periodical functions y 1= sin 2πx. . . ……………Period : T 1 = 1 y 2=sin 22πx ………… Period : T 2 = ? ? ? y 3 = Abs(sin 2πx) …… Period : T 3 = ? ? ? Note : sin 2 x = ½(1 - cos 2 x) = ½[1 - sin(2 x + π/2)] => T 2 = ? Beijing - 17/04/08 jml@ecole-alsacienne. org 14
General periodical functions in Physics g(t) = gmaxcos(wt + j) gmax = maximum value (positive) ) =3 Volts w = angular frequency = 200π , T = 0. 01 s. j = constant phase =100 (Rd) Beijing - 17/04/08 jml@ecole-alsacienne. org 15
That’s all folks … Mr. Lagouge will continue with many applications of these questions in Physics on Friday. Xiè ! Beijing - 17/04/08 jml@ecole-alsacienne. org 16
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