The smallest positive period of function in trigonometry is designated by f. It is characterized by the smallest value of positive number T, that is less its value T will not be the function period any more.
It is required to you
- - mathematical reference book.
Instruction
1. Pay attention that periodic function not always has the smallest positive period. So, for example, as the period of constant function there can be any number, so, at it can not be the smallest positive period. Meet as well changeable periodic functions which have no smallest positive period. However in most cases periodic functions nevertheless have the smallest positive period.
2. The smallest period of a sine is equal to 2?. Consider the proof of it on the example of the y=sin (x) function. Let T will be any period of a sine, in that case sin(a+T)=sin(a) at any value a. If a=?/2, turns out what sin (T+?/2) =sin (?/2) =1. However sin(x)=1 only when x=?/2+2? n where n represents an integer. From this it follows that T=2? n, so, the smallest positive value 2? n is 2?.
3. The smallest positive period of a cosine is equal to 2 too?. Consider the proof of it on the example of the y=cos (x) function. If T is any period of a cosine, then cos(a+T)=cos(a). In that case if a=0, cos(T)=cos(0)=1. So, the smallest positive value T at which 2 are cos(x)=1?.
4. Considering the fact that 2? – the period of a sine and cosine, the same value will be also the cotangent period and also a tangent, however not minimum as, as we know, the smallest positive period of a tangent and cotangent is equal?. You will be able to make sure of it, having reviewed the following example: the points corresponding to numbers (x) and (x +?) on a trigonometrical circle, have opposite arrangement. Distance from a point (x) to a point (x +2?) corresponds to a half of a circle. By definition of a tangent and cotangent of tg (x+?) =tgx, and ctg (x+?) =ctgx, so, the smallest positive period of a cotangent and tangent is equal?.