Question

What is the period of $\sin \left( {\pi x} \right)$?

Answer 1

First of all we have to know what is the periodicity of a function. So, the periodicity of a function is the period or angle after which the function is repeating its value. In the case of trigonometric functions, we know that these all are repetitive functions and they repeat themselves after a particular value. The value after which the function repeats itself among the domain is called its periodicity. So, let us see how to solve the function.

Complete step by step answer:
The sine function has period $2\pi$. This means that:
$\sin \left( {y + 2\pi } \right) = \sin y$
And if $0 < t < 2\pi$, then, we can say that,
$\sin \left( {y + t} \right) \ne \sin y,\forall y.$
Therefore, with $y = \pi x$, the function becomes,
$f\left( x \right) = \sin \left( {\pi x} \right) = \sin \left( {\pi x + 2\pi } \right) = \sin \left( {\pi \left( {x + 2} \right)} \right) = f\left( {x + 2} \right)$
And if $0 < s < 2$, the function becomes,
$f\left( {x + s} \right) = \sin \left( {\pi \left( {x + s} \right)} \right) = \sin \left( {\pi x + \pi x} \right) \ne \sin \left( {\pi x} \right)$
Because $0 < \pi s < 2\pi$.

Therefore, the period of $f$ is $T = 2$.

Note: The periodicity of a function can also be determined from the graph of a function. If we observe the graph of a function we can see the interval among which the graph repeats its curve or its trajectory. The domain among which it repeats itself will be its periodicity. Like the graph of sine function will be a wavy curve that repeats itself after the interval $2\pi$ over the domain of the curve which is the set of real numbers in case of sine function.