Advice 1: How to find the smallest positive period of the function

The smallest positive period of a function in trigonometry is denoted by f. It is characterized by the smallest value of a positive number T, that is, the smaller the value of T will not be a periodω of the function.
How to find the smallest positive period of the function
You will need
  • - mathematical reference book.
Instruction
1
Please note that the companynical function always has a least positive period. So, for example, as a periodand a constant function can be any number, and so it may not be the smallest positive period. There are also irregular period- diffraction functionswhich have no smallest positive period. However, in most cases, the smallest positive period from periodble 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 periodω 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 periodω 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 periodω 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.

Advice 2: How to find average salary

The calculation of the average wage of employees necessary for the calculation of benefits for temporary disability, business trip payments. The average professionals earnings is calculated on the basis of time actually worked, and depends on salary, allowances, bonuses specified in the staffing table.
How to find average salary
You will need
  • - staffing;
  • calculator;
  • legislation;
  • - production calendar;
  • - report card time attendance or certificate of completion.
Instruction
1
To make the calculation of the average wages of the employee, first determine the period for which it must be calculated. As a rule, this period acts 12 calendar months. But if the employee is working in the company less than a year, for example, for 10 months, then you need to find the average earnings for the time that the specialist performs their work functions.
2
Now determine the amount of wages which were actually accrued during the calculation period. To do this, use the payroll on which the employee was granted all the required payments. If you are unable to use these documents, monthly salary, bonuses, allowances and multiply by 12 (or the number of months that the employee is working in the company if it is executed in the company less than a year).
3
Calculate the average daily earnings. To do this, the amount of wages for the billing period, divide by the average number of days per month (currently it is 29.4). The result, divide by 12.
4
Then, determine the amount of time actually worked. To do this, use the sheet of accounting of working time. This document must complete the timekeeper, personnel clerk or another employee, which is written in job responsibilities.
5
The number of actual hours worked multiply the average daily earnings. The resulting amount is the average wage of a specialist within a year. The result, divide by 12. It will be the average monthly earnings. This calculation applies to employees who have payroll depends on the time actually worked.
6
Where the employee established piece-rate pay, the tariff rate (specified in the staffing table and the specific employment agreement) multiply by the number of produced products (use certificate of completion or other document in which it is fixed).
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