Question

The trigonometric function with a period 7 is:
A. \[\sin \dfrac{\pi }{7}\]
B. \[co\operatorname{s} \dfrac{{2\pi x}}{7}\]
C. \[\tan \dfrac{{2\pi x}}{7}\]
D. \[\cot \dfrac{{2\pi x}}{7}\]

Answer 1

Hint: Here the given question needs to be solved by finding the period for every function, which is equal to seven, here we know the period of the trigonometric function, and from which we can solve further. Period of a trigonometric value means, the angle after which the identity again repeats its value.
Formulae Used: Period of trigonometric function:
\[
   \Rightarrow \sin \theta = \cos \theta = 2\pi \\
   \Rightarrow \tan \theta = \cot \theta = \pi \\
 \]

Complete step by step answer:
Here we know the period for every identity and according to the question we need to solve for the period of seven, so assuming it we will solve further, on solving we get:
For sin function:
\[
   \Rightarrow \dfrac{{2\pi }}{{\left| a \right|}} = 7 \\
   \Rightarrow \left| a \right| = \dfrac{{2\pi }}{7} \\
   \Rightarrow \sin \left( {\dfrac{{2\pi }}{7}} \right) \\
 \]
For cosine function:
\[
   \Rightarrow \dfrac{{2\pi }}{{\left| a \right|}} = 7 \\
   \Rightarrow \left| a \right| = \dfrac{{2\pi }}{7} \\
   \Rightarrow \cos \left( {\dfrac{{2\pi }}{7}} \right) \\
 \]

For tan function:
\[
   \Rightarrow \dfrac{\pi }{{\left| a \right|}} = 7 \\
   \Rightarrow \left| a \right| = \dfrac{\pi }{7} \\
   \Rightarrow \tan \left( {\dfrac{\pi }{7}} \right) \\
 \]
For cot function:
\[
   \Rightarrow \dfrac{\pi }{{\left| a \right|}} = 7 \\
   \Rightarrow \left| a \right| = \dfrac{\pi }{7} \\
   \Rightarrow \cot \left( {\dfrac{\pi }{7}} \right) \\
 \]
Here we can see that cosine function is our required answer from the options.

So, the correct answer is “Option B”.

Note: In the given question, we need to remember the period of every trigonometric function, here we know the period of the function, so equate it with the required period as given in the question, so to find the answer. Here the period can be also remembered by the graph of the given function, when the curve reaches back to its past position then the desired range will be the period.