Let, #f(t)=g(t)+h(t), g(t)=sin(t/16), h(t)=cos(t/18).#
We know that #2pi# is the Principal Period of both #sin, &, cos#
functions (funs.).
#:. sinx=sin(x+2pi), AA x in RR.#
Replacing #x# by #(1/16t),# we have,
# sin(1/16x)=sin(1/16x+2pi)=sin(1/16(t+32pi)).#
#:. p_1=32pi# is a period of the fun. #g#.
Similarly, #p_2=36pi# is a period of the fun. #h#.
Here, it would be very important to note that, #p_1+p_2# is not
the period of the fun. #f=g+h.#
In fact, if #p# will be the period of #f#, if and only if,
#EE l, m in NN," such that, "lp_1=mp_2=p.........(ast)#
So, we have to find
#l,m in NN," such that, "l(32pi)=m(36pi), i.e.,#
#8l=9m.#
Taking, #l=9, m=8,# we have, from #(ast),#
#9(32pi)=8(36pi)=288pi=p,# as the period of the fun. #f#.
Enjoy Maths.!
