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In the nutshell, if “T” is the period of f(x), then period of function of the form given below id “T/|b|” :

a f ( b x + c ) + d ; a,b,c,d Z

Problem : What is the period of function :

f x = 3 + 2 sin { π x + 2 3 }

Solution : Rearranging, we have :

f x = 3 + 2 sin ( π 3 x + 2 3 )

The period of sine function is “ 2 π ”. Comparing with function form " a f ( b x + c ) + d ", magnitude of b i.e. |b| is π/3. Hence, period of the given function is :

T = T | b | = 2 π π 3 = 6

Modulus of trigonometric functions and periods

The graphs of modulus of a function are helpful to determine periods of modulus of trigonometric functions like |sinx|, |cosx|, |tanx| etc. We know that modulus operation on function converts negative function values to positive function values with equal magnitude. As such, we draw graph of modulus function by taking mirror image of the corresponding core graph in x-axis. The graphs of |sinx| and |cotx| are shown here :

Modulus of sine and cotangent functions

Period of function is π.
Period of function is π.

From the graphs, we observe that periods of |sinx| and |cotx| are π. Similarly, we find that periods of modulus of all six trigonometric functions are π.

Integral exponentiation of trigonometric function and periods

The periods of trigonometric functions which are raised to integral powers, depend on the nature of exponents. The periods of trigonometric exponentiations are different for even and odd powers. Following results with respect these exponentiated trigonometric functions are useful :

Functions sin n x , cos n x , cosec n x and sec n x are periodic on “R” with period “ π ” when “n” is even and “ 2 π ” when “n” is fraction or odd. On the other hand, Functions tan n x and cot n x are periodic on “R” with period “ π ” whether n is odd or even.

Problem : Find period of sin 2 x .

Solution : Using trigonometric identity,

sin 2 x = 1 + cos 2x 2

sin 2 x = 1 2 + cos 2x 2

Comparing with a f ( b x + c ) + d , the magnitude of “b” i.e. |b| is 2. The period of cosine is 2π. Hence, period of sin 2 x is :

T = 2 = π

Problem : Find period of function :

f x = sin 3 x

Writing identity for " sin 3 x ", we have :

f x = sin 3 x = 3 sin x sin 3 x 4 = 3 4 sin x 3 4 sin 3 x

We know that period of “ag(x)” is same as that of “g(x)”. The period of first term of “f(x)”, therefore, is equal to the period of “sinx”. Now, period of “sinx” is “2π”. Hence,

T 1 = 2 π

We also know that period of g(ax+b) is equal to the period of g(x), divided by “|a|”. The period of second term of “f(x)”, therefore, is equal to the period of “sinx”, divided by “3”. Now, period of “sinx” is “2π”. Hence,

T 2 = 2 π 3

Applying LCM rule,

T = LCM of 2 π and 2 π HCF of 1 and 3 = 1 = π

Lcm rule for periodicity

When two periodic functions are added or subtracted, the resulting function is also a periodic function. The resulting function is periodic when two individual periodic functions being added or subtracted repeat simultaneously. Consider a function,

f(x) = sinx + sin x 2

The period of sinx is 2π, whereas period of sinx/2 is 4π. The function f(x), therefore, repeats after 4π, which is equal to LCM of (least common multiplier) of the two periods. It is evident from the graph also.

Graphs of sine functions

Two functions repeat values after 4π.

Questions & Answers

what is the meaning of function in economics
Effah Reply
Pls, I need more explanation on price Elasticity of Supply
Isaac Reply
Is the degree to the degree of responsiveness of a change in quantity supplied of goods to a change in price
Afran
Discuss the short-term and long-term balance positions of the firm in the monopoly market?
Rabindranath Reply
hey
Soumya
hi
Mitiku
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Mitiku
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Soumya
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Mitiku
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Mitiku
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Rabindranath
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Isaac
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Afran
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Afran
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Anteyi Reply
A firm is a business entity which engages in the production of goods and aimed at making profit.
Avuwada
What is autarky in Economics.
Avuwada
what is choice
Tia Reply
So how is the perfect competition different from others
Rev Reply
what is choice
Tia
please what type of commodity is 1.Beaf 2.Suagr 3.Bread
Alfred Reply
1
Naziru
what is the difference between short run and long run?
Ukpen Reply
It just depends on how far you would like to run!!!🤣🤣🤣
Anna
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Ukpen
pls is question from which subject or which course
Ada
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Ukpen
This place is meant to be for serious educational matters n not playing ground so pls let's make it a serious place.
Docky
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Docky
Okay and I was being serous
Anna
The short run is a period of time in which the quantity of at least one inputs is fixed...
Anna
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Anna
Elacisity
salihu
Meaning of economics
Suraj Reply
It will creates rooms for an effective demands.
Chinedum Reply
different between production and supply
babsnof
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Suraj
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eshita
What is the economic?
Suraj
Economics is a science which study human behavior as a relationship between ends and scarce means which has an alternative use.
Mr
what is supply
babsnof
what is different between demand and supply
Debless Reply
Demand refers to the quantity of products that consumers are willing to purchase at various prices per time while Supply has to do with the quantity of products suppliers are willing to supply at various prices per time. find the difference in between
Saye
Please what are the effects of rationing Effect of black market Effects of hoarding
Atty Reply
monoply is amarket structure charecrized by asingle seller and produce a unique product in the market
Cali Reply
yes
Niraj
I want to know wen does the demand curve shift to the right
Nana
demand curve shifts to the right when there's an increase in price of a substitute or increase in income
kin
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kin
what are the factor that change the curve right
Nana
explain the law of supply in simple .....
freshwater
the Law of supply: states that all factor being equal, when the price of a particular goods increase the supply will also increase, as it decreases the supply will also decrease
kin
@Nana the factor that changes or shift the d demand curve to the right is 1) the increase in price of a substitute good or commodity 2) increase in income
kin
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kin
different between bill of exchange n treasure bill
Nana
yes
Ada
so would you tell me what means an apportunity cost plz?
Cali
what is true cost
Akiti
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Cali
In a simple term, it is an Alternative foregone.
Sule
opportunity cost is the next best value of a scale of preference
Akiti
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Nelly
opportunity cost: is a forgone alternative
kin
Monopoly is where is one producer produces a given product with no close substitute
James
what is income effect?
Qwecou Reply
if you borrow $5000 to buy a car at 12 percent compounded monthly to be repaid over the next 4 year what is monthly payment
Nitish Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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