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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
Discuss the short-term and long-term balance positions of the firm in the monopoly market?
Rabindranath Reply
how are you?
can you tell how can i economics honurs(BSC) in reputed college?
through hard study and performing well than expected from you
what should i prepare for it?
prepare first, in psychologically as well as potentially to sacrifice what's expected from you, when I say this I mean that you have to be ready, for every thing and to accept failure as a good and you need to change them to potential for achievement of ur goals
parna kya hai behencho?
Hello, dear what's up?
good morning
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what is firms
Anteyi Reply
A firm is a business entity which engages in the production of goods and aimed at making profit.
What is autarky in Economics.
what is choice
Tia Reply
So how is the perfect competition different from others
Rev Reply
what is choice
please what type of commodity is 1.Beaf 2.Suagr 3.Bread
Alfred Reply
what is the difference between short run and long run?
Ukpen Reply
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Is there an economics expert here?
Okay and I was being serous
The short run is a period of time in which the quantity of at least one inputs is fixed...
that is the answer that I found online and in my text book
Meaning of economics
Suraj Reply
It will creates rooms for an effective demands.
Chinedum Reply
different between production and supply
What is the economic?
Economics is a science which study human behavior as a relationship between ends and scarce means which has an alternative use.
what is supply
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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
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
I want to know wen does the demand curve shift to the right
demand curve shifts to the right when there's an increase in price of a substitute or increase in income
ask me anything in economics, I promise to try and do justice to the question, you can send me an email or message, I will answer
what are the factor that change the curve right
explain the law of supply in simple .....
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
@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
you can send your questions I am Comr. Kin chukwuebuka
different between bill of exchange n treasure bill
so would you tell me what means an apportunity cost plz?
what is true cost
your question isn't correct naadi
define an apportunity cost?
orukpe ,is my question whats wrong or u dont know anything?
In a simple term, it is an Alternative foregone.
opportunity cost is the next best value of a scale of preference
Both of you are not correct.
opportunity cost: is a forgone alternative
Monopoly is where is one producer produces a given product with no close substitute
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
Sneha Reply

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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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