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

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 $2\pi$ .