Question
Which of the following functions has period $$2\pi \,?$$
A.
$$y = \sin \left( {2\pi \,t + \frac{\pi }{3}} \right) + 2\sin \left( {3\pi \,t + \frac{\pi }{4}} \right) + 3\sin 5\pi \,t$$
B.
$$y = \sin \frac{\pi }{3}t + \sin \frac{\pi }{4}t$$
C.
$$y = \sin t + \cos 2t$$
D.
None of these
Answer :
$$y = \sin t + \cos 2t$$
Solution :
We have two functions $$f\left( x \right)$$ and $$g\left( x \right)$$ have periods $$T_1$$ and $$T_2$$ respectively, then each of $$f\left( x \right) \pm g\left( x \right);f\left( x \right) \cdot g\left( x \right);\frac{{f\left( x \right)}}{{g\left( x \right)}},$$ provided $$g\left( x \right) = 0$$ has period equal to the LCM of $$T_1$$ and $$T_2 .$$
Now, we know that $$\sin x$$ or $$\cos x$$ has period $$2\pi .$$
Hence, period of $$y = \sin t + \cos 2t{\text{ is }}2\pi .$$