Math 3 Unit 6, Trigonometry L04: Amplitude and Period of Sine and Cosine AND Translations of Sine and Cosine Functions WIMD: What I must do: I will find the amplitude and period from a graph of the sine and cosine function I will create the equation for sine or cosine graph with a given amplitude and period. I will identify the phase shift and vertical translation for sine and cosine functions. I will write the equation given the amplitude, period, phase shift, and vertical translation. Key Words: phase shift amplitude period midline Sep 25 9:06 AM What is a Sine Function Graph? U4 L2 Relate Circle to Sine Activity.pdf Dec 16 8:14 AM
Amplitude sine midline cosine The amplitude is the distance from the midline. y= ±Asin( θ), y= ±Acos( θ) A is the amplitude. Mar 17 2:20 AM Vertical Translation (i.e. Move the midline up or down) What is the midline of y=cos(x)? What is the midline of y=cos(x)+3? y = 0 y = 3 1 Answer?? Mar 17 11:21 PM
Periodicity f(x) = f(x + T) T is your period if this is true for all x Dec 9 12:33 PM Suppose: f(θ)= sin(kθ) Note: Domain is radians Note: Domain is radians sin(kθ)= sin(kθ + 2π) ; 2π period of sine function. sin(kθ) = sin[k(θ + 2π/k)] (kθ) = k(θ + 2π/k) θ = θ + 2π/k f(θ) = f(θ + 2π/k) ; by substitution. f(θ) = f(θ + T) ;by definition of periodicity T = 2π/k ; factor out the k. ; inverse sine of both sides. ; divide by k. Note: k = (2 π/τ ); T is the period in radians Mar 17 10:21 PM
Alternatively: Suppose f(t) = sin( ωt) ; ω= θ / t = sin( ωt+2 π) = sin[ ω(t+2 π/ω) ] = sin(ωt) (ωt) = ω(t+2π/ω) t = t+2π/ω f(t) = f(t+2 π/ω ) If f(t) is a periodic function, then T=2 π/ω ω = 2π/ T Angular Velocity Radians/sec Mar 20 5:13 PM Note: Domain is time Frequency The number of cycles per unit of time. Domain is time FYI: Mar 20 5:25 PM
What is the period of sin(20θ)? 2 Answer? 3 decimals Τ 0.314 Mar 14 4:06 PM Relate to translations vertical: >1 stretch <1 compress horizontal: >1 compress <1 stretch vertical shift: >0 moves up <0 moves down ±A is the amplitude period, T = 2π / k horizontal phase shift Mar 18 12:12 AM
I will create the equation for sine or cosine graph with a given amplitude and period. Mar 17 10:32 PM Steps to graph Sine or Cosine: Determine the vertical shift, and draw the midline. Determine the amplitude. Draw in dashed lines to show maximum or minimum values. Determine phase shift, then translate Determine the period, then graph a dashed version of the sine/cosine. (Use dashed curve) Oct 23 10:32 AM
Sketch 2sin( θ) 1 1 sine( θ) 0 1 Amplitude: Vert Shift: Midline: Period: Mar 17 2:20 AM How do I know when to choose the sine or cosine function? Ans: Examine your initial conditions At f(0) = 0 At f(0) = max/min Mar 20 4:17 PM
( ) Mar 18 12:35 AM The motion of a weight on a certain kind of spring can be described by a modified trigonometric function. At time 0, Carrie pushes the weight upward 3 inches from its equilibrium point and releases it. She finds that the weight returns to the point three inches above the equilibrium point after 2 seconds. T = 2 sec. A = 3 inches y = 3cos(πt) Mar 20 5:05 PM
Find the period, amplitude, midline, vertical shift and horizontal phase shift of the function. A = 2/3 V.S. = up 7.4 T = π/4 phase shift = left 15 π/8 midline: y = 7.4 Oct 21 9:54 AM EXAMPLE Given the equation Identify the amplitude, period, vertical shift, and phase shift. Then graph the function. 4 3 2 1 π 2π 3π 4π 5π 6π Mar 18 12:47 AM
Glencoe Quizzes Access Code: E7BE47BB9B Homework #49 52, 60 #31 35, 43, 44 p. 374 #49 52, 60 Mar 21 5:10 AM
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Attachments Prentice Hall Algebra On line Resources 06.1 Trig Circle Summary.doc U4 L2 Relate Circle to Sine Activity.pdf