You will need

- - mathematical reference book.

Instruction

1

Please note that

**the company**nical function always has a least positive**period**. So, for example, as a**period**and a constant**function**can be any number, and so it may not be the smallest positive**period**. There are also irregular**period**- diffraction**functions**which have no smallest positive**period**. However, in most cases, the smallest positive**period**from**period**ble functions are still there.2

The smallest

**period**of sine is 2?. Consider the proof on the example**of the function**y=sin(x). Let T be an arbitrary**period**ω of the sine, in this case sin(a+T)=sin(a) with any value of a. If a=?/2, it turns out that sin(T+?/2)=sin(?/2)=1. However, sin(x)=1 only if x=?/2+2?n, where n is an integer. It follows that T=2?n, and hence the smallest positive value of 2?n is 2?.3

The smallest positive

**period**of the cosine is also equal to 2?. Consider the proof on the example**of the function**y=cos(x). If T is an arbitrary**period**ω of the cosine, cos(a+T)=cos(a). If a=0, cos(T)=cos(0)=1. In view of this, the smallest positive value of T where cos(x)=1, 2?.4

Given the fact that 2? –

**period**of sine and cosine, it's value will be and**the period**ω of the cotangent and tangent, but not minimal, because, as you know, the smallest positive**period**of tangent and cotangent is equal to ?. To make sure this will the following example: the points corresponding to the numbers (x) and (x+?) on the trigonometric circle, have a diametrically opposite location. The distance from the point (x) to the point (x+2?) corresponds to half the circumference. By definition of tangent and cotangent tg(x+?)=tgx, ctg(x+?)=ctgx and, therefore, the smallest positive**period**of the cotangent and tangent equal ?.Note

Not to be confused of the function y=cos(x) and y=sin(x) having the same period, these functions are represented differently.

Useful advice

For greater clarity, draw a trigonometric function, which calculates the smallest positive period.