Question

The period of the function \[f\left( x \right) = \left| {\sin x} \right| + \left| {\cos x} \right|\] is,
1.\[\pi \]
2.\[\dfrac{\pi }{2}\]
3.\[2\pi \]
4.None of these

Answer 1

Hint: In the above solution, we are given a combination of modulus function and trigonometric function of the variable \[x\] . The given function is written as \[f\left( x \right) = \left| {\sin x} \right| + \left| {\cos x} \right|\] . We need to determine the period of the above given function of \[x\] . In order to approach the solution, first we need to consider the mathematical definition of the time period of a function. When we have a function such that \[f\left( x \right) = f\left( {x + {\rm T}} \right)\] , then we say that the period of this function \[f\left( x \right)\] is \[T\] . Hence, we have find a time period \[T\] such that it gives us the equation:
\[ \Rightarrow f\left( x \right) = \left| {\sin x} \right| + \left| {\cos x} \right| = f\left( {x + T} \right) = \left| {\sin x + T} \right| + \left| {\cos x + T} \right|\]

Complete answer:
Given that, a function of \[x\] which is written as,
\[ \Rightarrow f\left( x \right) = \left| {\sin x} \right| + \left| {\cos x} \right|\]
Here, we know that the range of function \[\left| {\sin x} \right|\] is \[\left[ {0,1} \right]\] ,
Whereas, the range of the function \[\left| {\cos x} \right|\] is also \[\left[ {0,1} \right]\] .
Hence, the period of both the functions \[\left| {\sin x} \right|\] and \[\left| {\cos x} \right|\] is equal to \[\pi \] .
So when we consider \[T = \dfrac{\pi }{2}\] , then we have the function \[f\left( {x + T} \right)\] as
\[ \Rightarrow f\left( {x + \dfrac{\pi }{2}} \right) = \left| {\sin \left( {\dfrac{\pi }{2} + x} \right)} \right| + \left| {\cos \left( {\dfrac{\pi }{2} + x} \right)} \right|\]
That gives us the equation as,,
\[ \Rightarrow f\left( {x + \dfrac{\pi }{2}} \right) = \left| { - \cos x} \right| + \left| {\sin x} \right|\]
That can also be written as,
\[ \Rightarrow f\left( {x + \dfrac{\pi }{2}} \right) = \left| {\cos x} \right| + \left| {\sin x} \right|\]
That gives us,
\[ \Rightarrow f\left( {x + \dfrac{\pi }{2}} \right) = \left| {\sin x} \right| + \left| {\cos x} \right|\]
Hence, we have
\[ \Rightarrow f\left( {x + \dfrac{\pi }{2}} \right) = f\left( x \right)\]
Therefore, the period of the function \[f\left( x \right)\] is \[T = \dfrac{\pi }{2}\] .

So the correct option is (2).

Note:
The distance between the repetition of any function is known as the period of the function. For a trigonometric function, the length of one complete cycle is called a period. For any trigonometric graph function, we can take \[x = 0\] as the starting point. The time period of the sine function and the cosine function is \[2\pi \] .