Amplitude, Period and Frequency

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Amplitude and Period of Sine and Cosine

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Key Questions

  • If #f(x)=asin(bx)# or #g(x)=acos(bx)#, then their amplitudes are #|a|#, and the periods are #{2pi}/|b|#.


    I hope that this was helpful.

  • Frequency and period are related inversely. A period #P# is related to the frequency #f#
    # P = 1/f#

    Something that repeats once per second has a period of 1 s. It also have a frequency of # 1/s#. One cycle per second is given a special name Hertz (Hz). You may also say that it has a frequency of 1 Hz.

    A sin function repeats regularly. Its frequency (and period) can be determined when written in this form:

    #y(t) = sin(2pi f t)#

  • The coefficient of the angle is key to both; it is the number of cycles in #2pi# radians (or #360# degrees) and hence represents the frequency on #2pi#, which in graphing is commonly referred to as just the "frequency," while simultaneously being the divisor of #2pi# radians (or #360# degrees) that results in the length to complete one cycle, which is called the period (my one behemoth sentence answer).

    In short, if you have a trigonometric function in the form:
    # Asin(B(x-C)) + D # or
    # Acos(B(x-C)) + D#

    #Period = (2pi)/(B)#

    #Frequency = (1)/(Period) #

    For tangent: #Atan(B(x-C))+D #, the period would be #(pi)/B#

    Let's try an example:

    graph{3sin(2x) [-16.02, 16.02, -8.35, 8.34]}

    #y = 3 sin 2x#

    In short terms, the period is # (2pi)/2 = pi #
    The frequency is #1/pi#

    In longer in-depth terms, it has a frequency (on #2pi# or #360# degrees) of #2#, meaning that there will be #2# complete cycles from #0# to #2pi# (or #360# degrees), or any domain with that width (like #-pi# to #pi#).

    The period, or length of each cycle can be found by dividing #2pi# or #360# by that coefficient revealing that in the above example, each complete cycle is #pi# or #180# degrees in width.

    As a side note: in real world applications (word problems) frequency is actually the reciprocal of period and so is the angles coefficient divided by #2pi# (or possibly #360#), or# 2/2pi# (#2/360#) in the above example. This is because period is time or distance per cycle, while frequency should be cycles per unit time (not per #2pi# units of time because the world doesn't count like that--we want to know how many revolutions per second our tires are making, not how many revolutions in #6.28# seconds, usually.)

    Thanks for the question.

  • #"(Amplitude)"=1/2["(Highest Value)"-"(Lowest Value)"]#


    graph{4sinx [-11.25, 11.25, -5.62, 5.625]}

    In this sine wave the highest value is #4# and the lowest is #-4#

    So the maximum deflection from the middle is #4#k.

    This is called the amplitude

    If the middle value is different from #0# then the story still holds
    graph{2+4sinx [-16.02, 16.01, -8, 8.01]}

    You see the highest value is 6 and the lowest is -2,
    The amplitude is still #1/2 (6- -2)=1/2 *8=4#

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