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:
\[\begin{align}
  & tan\left( 3x+5 \right)\,\,\,is\,\, \\
 & \dfrac{\text{periodicity}\;\text{of}\;\tan \left( x \right)}{3}=\dfrac{\pi }{3} \\
\end{align}\]
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.