- #1
geniusprahar_21
- 28
- 0
Is there a continuous periodic function which is not trigonometric. if yes, what?
Yes there are very many. Define a continuous function on [a,b] where f(a)=f(b) then define f outside of [a,b] so that f(x+(b-a))=f(x). A simple example that is not trigonometric (even though it looks like it is) is Arccos(cos(x)).geniusprahar_21 said:Is there a continuous periodic function which is not trigonometric. if yes, what?
...f(x) = Arccos(cos(x)) = x...
That is right Arccos(cos(x))=x on [0,pi], it is also periodic with period pi, so it repeats all those values. I use Arccos with the capital A to make clear that I am using the principle value of Arccos not just any value that gives the needed value. This is a general way to write periodic functions. let f(x) be diffined and continuous on [a,b] with f(a)=f(b) thenMuzza said:No, the inverse cosine function returns values in a specific interval (which I can't remember atm), so you can't have arccos(cos(x)) = x for all x.
Every function is man-made. Not every function is "man-made".HallsofIvy said:I don't know any functions that aren't "man-made"!
A periodic function is a function that repeats its values at regular intervals, called the period. This means that the same output values occur at regular intervals as the input values change.
Yes, a function can be periodic without involving trigonometric functions. A simple example is the sawtooth wave function, which repeats its values in a linear pattern. Other examples include square waves, triangle waves, and step functions.
To determine if a function is periodic, we need to find a constant value, called the period, which when added to or subtracted from the input value, gives the same output value. If such a constant exists, then the function is periodic. Otherwise, the function is not periodic.
Yes, there are many non-trigonometric functions that can model periodic phenomena. For example, the logistic function can be used to model population growth, and the sine logistic function can be used to model the spread of infectious diseases. These functions exhibit periodic behavior without involving trigonometric functions.
Yes, there are benefits to using non-trigonometric functions to model periodic phenomena. Non-trigonometric functions can provide a more accurate representation of real-world phenomena, as they can capture more complex and irregular periodic patterns. Additionally, non-trigonometric functions may be easier to work with and analyze mathematically compared to trigonometric functions.