Question

The period of $$\sin\dfrac{\mathrm\pi\left[\mathrm x\right]}{12}+\cos\dfrac{\mathrm\pi\left[\mathrm x\right]}4+\tan\dfrac{\mathrm\pi\left[\mathrm x\right]}3$$ where [x] represents the greatest integer less than or equal to x is-
A)12
B)4
C)3
D)24

Answer 1

Hint: One should have the knowledge about the period of trigonometric functions. Also, we should know that when different functions with different periods are added then the resultant period is their LCM.

Complete step-by-step answer:
We know that sine function has a period of $$2\mathrm\pi$$, so the period of the first term is-

$$\dfrac{2\mathrm\pi}{\left({\displaystyle\dfrac{\mathrm\pi}{12}}\right)}=24$$

The period of cosine function is $$2\mathrm\pi$$, so the period of the second term is-

$$\dfrac{2\mathrm\pi}{\left({\displaystyle\dfrac{\mathrm\pi}4}\right)}=8$$

The period of tangent function is $$\mathrm\pi$$, so the period of the third term is-

$$\dfrac{\mathrm\pi}{\left({\displaystyle\dfrac{\mathrm\pi}3}\right)}=3$$

Hence, the period of the complete function is equal to the periods of the three terms-

Period = LCM(24, 8, 3)

Period = 24

Hence, the correct option is D. 24


Note: Here, we do not consider the period of the greatest integer function as its period is infinite. Also, when finding the period of the function, divide the original period by the coefficient of x to get the correct period.