Amplitude, Period and Frequency
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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(xC)) + D # or
# Acos(B(xC)) + D# #Period = (2pi)/(B)# #Frequency = (1)/(Period) # For tangent:
#Atan(B(xC))+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 indepth 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 thatwe 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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