What is the period of #sin(3*x)+ sin(x/(2))#?

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Answer 1 · 2016-07-17T06:59:22

The Prin. Prd. of the given fun. is #4pi#.

Let #f(x)=sin3x+sin(x/2)=g(x)+h(x)#, say.

We know that the Principal Period of #sin# fun. is #2pi#. This

means that, #AA theta, sin(theta+2pi)=sintheta#

#rArr sin3x=sin(3x+2pi)=sin(3(x+2pi/3))#

#rArr g(x)=g(x+2pi/3)#.

Hence, the Prin. Prd. of the fun. #g# is #2pi/3=p_1#, say.

On the same lines, we can show that, the Prin. Prd. of the fun #h# is

#(2pi)/(1/2)=4pi=p_2#, say.

It should be noted here that, for a fun. #F=G+H#, where,

#G and H# are periodic funs. with Prin. Prds. #P_1 & P_2,# resp.,

it is not at all necessary that the fun. #F# be periodic .

However, #F# will be so, with Prin. Prd. #p#, if we can find,

#l,m in NN#, such that, #l*P_1=m*P_2=p#.

So, let us suppose that, in our case, for some #l,m in NN,#

#l*p_1=m*p_2=p.............(1)#

#rArr l*(2pi)/3=m*4pi rArr l=6m#

So, by taking, #l=6, and m=1#, we have, from #(1)#,

#6*(2pi/3)=1*(4pi)=p=4pi#

Hence, the Prin. Prd. of the given fun. is #4pi#.