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Learn about the construction, properties, and applications of sine and cosine functions, including period, maximum, minimum, symmetry, translations, and relationships between them. Discover how these functions are used in various fields, such as physics and signal processing. Gain insights into Fourier's theorem and general periodic functions. Join Mr. Lagouge in exploring the fascinating world of trigonometric functions.
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Sine & Cosine functions jml@ecole-alsacienne.org
Construction of the Sine function jml@ecole-alsacienne.org
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
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
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
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
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
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
Properties of the sine function : • 8. For values of x close to 0, sin x ≈ x jml@ecole-alsacienne.org
Transfert from Sine to Cosine Cos x = Sin(x + π/2) Sin x = Cos(π/2 - x) jml@ecole-alsacienne.org
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
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
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
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
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
That’s all folks … Mr. Lagouge will continue with many applications of these questions in Physics on Friday. Xiè Xiè ! jml@ecole-alsacienne.org
