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Sine & Cosine functions

Sine & Cosine functions. Construction of the Sine function. 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)

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Sine & Cosine functions

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  1. Sine & Cosine functions jml@ecole-alsacienne.org

  2. Construction of the Sine function jml@ecole-alsacienne.org

  3. 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) jml@ecole-alsacienne.org

  4. 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 jml@ecole-alsacienne.org

  5. 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 jml@ecole-alsacienne.org

  6. 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. jml@ecole-alsacienne.org

  7. Properties of the sine function : • 5. Unchanged by any translation of k.2π along the Ox axis Sin(x + k2π) = Sin(x) That is to justify the construction of the curve by copying any part of length = 2π as many times a we can. jml@ecole-alsacienne.org

  8. 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…) jml@ecole-alsacienne.org

  9. Properties of the sine function : • 8. For values of x close to 0, sin x ≈ x jml@ecole-alsacienne.org

  10. Transfert from Sine to Cosine Cos x = Sin(x + π/2) Sin x = Cos(π/2 - x) jml@ecole-alsacienne.org

  11. General Sine functions f(x) = Asin(ax + b) A=amplitude a =2π/T , T = period =2π /a. PROOF ? b =constant phase (ax +b) = phase jml@ecole-alsacienne.org

  12. Combinations of sine functions y1=2.sin(2π/3)x ...………..Period : T1 = 3 y2=3.sin(πx) …………….Period : T2 = 2 y3 =y1 + y2 = 2.sin(2πx/3) + 3sin(πx) Prove that the Period T3 = LCM (T1 ; T2) = 2 x 3 = 6 jml@ecole-alsacienne.org

  13. Fourier’s theorem The sum of periodic functions is also a periodic function. Any periodical function can be written as the sum of a series of sine functions. Even a square signal … jml@ecole-alsacienne.org

  14. Other kind of periodical functions y1= sin 2πx...……………Period : T1 = 1 y2=sin22πx ………… Period : T2 = ??? y3 = Abs(sin 2πx)…… Period : T3 = ??? Note : sin2x = ½(1 - cos 2x) = ½[1 - sin(2x + π/2)] => T2 = ? jml@ecole-alsacienne.org

  15. 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) jml@ecole-alsacienne.org

  16. That’s all folks … Mr. Lagouge will continue with many applications of these questions in Physics on Friday. Xiè Xiè ! jml@ecole-alsacienne.org

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