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Q.
The period of the function $\sin\left(\frac{2x}{3}\right)+\sin\left(\frac{3x}{2}\right) $ is
Trigonometric Functions
Solution:
As $\sin \frac{2x}{3} = \sin\left(2\pi + \frac{2x}{3}\right)$
$ =\sin\left(\frac{2}{3} \left(3\pi+x\right)\right), $ therefore , period of $ \sin \frac{2x}{3}$ is $3\pi$ and also
$ \sin \frac{3x}{2} =\sin\left(2\pi + \frac{3x}{2}\right) = \sin\left(\frac{3}{2} \left(\frac{4 \pi}{3} + x\right)\right) $ therefore, period of $ \sin \, \frac{3x}{2}$ is $ \frac{4\pi}{3} $.
Hence penod of f(x) is L.C.M. of $3 \pi$ and $\frac{4 \pi}{3}, $ i.e.,$12 \pi $.
($\because$ set of multiples of $3 \pi={3 \pi,6 \pi, 9 \pi, 12 \pi, .....)}$ and set of multiplies of $\frac{4 \pi}{3}$
$ =\left\{\frac{4\pi}{3} , \frac{8\pi}{3} , \frac{12\pi}{3} , \frac{16\pi}{3}m,...\right\}$