Question

The period of the function $tan\left( 3x+5 \right)$is
A. $\dfrac{2\pi }{3}$
B. $\dfrac{\pi }{6}$
C. $\dfrac{\pi }{3}$
D. None of these

Answer 1

Hint: Use the concept that the period of the function of the form $a\times \tan \left( bx\;+\;c \right)\;+\;d\,\,is\,\,\dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{|b|}$. Also, use the concept that the period of $\tan \left( x \right)\,\,\;\text{is}\,\,\pi$. Now, we just need to just compare the function given with the standard form as shown in the formula and then use the above formula to get the required period.

Complete step-by-step answer:
In the question, we have to find the period of the function $tan\left( 3x+5 \right)$. Now, it is known that if the function is of the form $a\times \tan \left( bx\;+\;c \right)\;+\;d$, then the period of that will be given by $\dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{|b|}$. So here, we can compare the given function $tan\left( 3x+5 \right)$with the expression $a\times \tan \left( bx\;+\;c \right)\;+\;d$ and we see that $a=1,\text{ }d=0,\text{ }b=3\text{ }and\text{ }c=5$. So, the period will be then found directly by using the above formula.
So here, we will see that the period of the function tan x is $\pi$, as this is the interval cafter which the function tan x is repeating itself. Now, this means that after every $\pi$ interval, we will have exactly the same behaviour of the function tan x.
Now, applying the formula that period of $a\times \tan \left( bx\;+\;c \right)\;+\;d=\dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{|b|}$, so the period of $tan\left( 3x+5 \right)\,\,\,is\,\,\dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{3}$as we just have seen that b=3.
Also, we have seen that the period of tan x is $\pi$. So, finally, the required period is s follows:

& tan\left( 3x+5 \right)\,\,\,is\,\, \\\ & \dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{3}=\dfrac{\pi }{3} \\\ \end{aligned}$$ So, the correct answer is option C which is $$\dfrac{\pi }{3}$$. Note: When we are finding the period of tan x, then it will not be $$2\pi $$, as we have the period of sin x and cos x. But here the period of tan x is just $$\pi $$, which is important to take care about.