## What are Periodic Functions?
Periodic functions are applied to study signals and waves in electrical and electronic systems, vibrations in mechanical and civil engineering systems, waves in physics and wireless systems and has many other applications.
Example 1
- \( \sin(x + 2\pi ) = \sin(x) \) , the period of \( \sin(x) \) is equal to \( P = 2\pi \)
The graph of \( \sin(x) \) is shown below with one cycle, in red, whose length over the x axis is equal to one period P given by: \( P = 2 \pi - 0 = 2 \pi \)
- \( \cos(x + 2\pi ) = \cos(x) \) , the period of \( \cos(x) \) is equal to \( P = 2\pi \)
- \( \sec(x + 2\pi ) = \sec(x) \) , the period of \( \sec(x) \) is equal to \( P = 2\pi \)
- \( \csc(x + 2\pi ) = \csc(x) \) , the period of \( \csc(x) \) is equal to \( P = 2\pi \)
- \( \tan(x + \pi ) = \tan(x) \) , the period of \( \tan(x) \) is equal to \( P = \pi \)
The graph of \( \tan(x) \) is shown below with one cycle, in red, whose length over the x axis is equal to one period P given by: \( P = \dfrac{\pi}{2} - (-\dfrac{\pi}{2} ) = \pi \)
- \( \cot(x + \pi ) = \cot(x) \) , the period of \( \cot(x) \) is equal to \( P = \pi \)
## Period of Transformed Functions1) If \( P \) is the period of \( f(x) \), then the period of \( A f(b x + c ) + D \) is given by \( \dfrac{P}{|b|} \)2) If \( P \) is the period of \( f(x) \), then \( f(x + n P) = f(x) \) , for \( n \) integer
Example 2
- \( f(x) = \sin(0.5 x) \)
- \( g(x) = \tan(2 x + \pi/6) \)
- \( h(x) = \cos(-(2/3) x - \pi) \)
- \( j(x) = \sec(\pi x - 2) \)
- \( k(x) = \cot(-(2\pi/3) x) \)
Solution to Example 2
- The period of \( \sin(x) \) is \( 2\pi \). We use the above formula to find the period of \( f(x) = \sin(0.5 x) \) as follows: \( \dfrac{2\pi }{|0.5|} = 4\pi\)
- The period of \( \tan(x) \) is \( \pi \), hence the period of \( g(x) = \tan(2 x + \pi/6) \) is equal to \( \dfrac{\pi }{|2|} \)
- The period of \( \cos(x) \) is \( 2\pi \), hence the period of \( h(x) = \cos(-(2/3) x - \pi) \) is equal to \( \dfrac{2\pi }{|-2/3|} = 3\pi\)
- The period of \( \sec(x) \) is \( 2\pi \) and the period of\( j(x) = \sec(\pi x - 2) \) is given by \( \dfrac{2\pi }{|\pi|} = 2\)
- The period of \( \cot(x) \) is \( \pi \) and the period of \( k(x) = \cot(-(2\pi/3) x) \) is given by \( \dfrac{\pi }{|-2\pi/3|} = 3/2\)
## More Examples with SolutionsFind the period of each of the functions given below
Example 3
- \( f(x) = \sin(x) \cos(x) \)
- \( g(x) = \sin^2(x) \)
- \( h(x) = \cos(x) + \sin(x) \)
Solution to Example 3
- Use the identity \( \sin(2x) = 2 \sin(x) \cos(x) \) to write \( f(x) = \sin(x) \cos(x) = (1/2) \sin(2x) \), hence the period of \( f(x) \) is given by \( \dfrac{2\pi }{|2|} = \pi\)
- Use the identity \( cos(2x) = 1 - 2 \sin^2(x) \) to write \( g(x) = \sin^2(x) = \dfrac{1}{2} cos(2x)+ \dfrac{1}{2} \), hence the period of \( g(x) \) is given by \( \dfrac{2\pi }{|2|} = \pi \)
- Use trigonometric identity of a sum to expand \( \sin(x + \pi/4) = \sin(x) \cos(\pi/4) + \cos(x) \sin(\pi/4) = \dfrac{\sqrt 2}{2}{\sin(x) + \dfrac{\sqrt 2} \cos(x)} \) to rewrite \( h(x) \) as
\( h(x) = {\sin(x) + \cos(x)} = \dfrac{2}{\sqrt 2}{\sin(x + \pi/4)} \) and calculate the period of \( h(x) \) as: \( \dfrac{2\pi }{|1|} = 2\pi \)
## More References and Links to Functionsfunctions |