Trigonometrical functions are periodical, that is repeat through a certain period. Thanks to it it is enough to investigate function on this interval and to extend the found properties to all other periods.
Instruction
1. If you were given simple expression at which there is only one trigonometrical function (sin, cos, tg, ctg, sec, cosec), and the corner in function is not increased by any number, and she is not built in any degree – use definition. For the expressions containing sin, cos, sec of cosec safely put the period 2P and if in the equation there is tg, ctg – that P. Naprimer, for function at =2 sinx +5 the period will be equal 2P.
2. If the corner x under the sign of trigonometrical function is increased by any number, then to find the period of this function, divide the standard period into this number. For example, to you function at = is given to sin of the 5th. The standard period for a sine – 2P, having divided it into 5, you receive 2P/5 - it is and there is a required period of this expression.
3. To find the period of the trigonometrical function built in degree, estimate parity of degree. For even degree reduce the standard period twice. For example, if you were given function at =3 cos^2х, then the standard period 2P will decrease twice, thus, the period will be equal to P. Obratita attention, the functions tg, ctg in any degree are periodical the Item.
4. If you were given the equation, the containing work or private two trigonometrical functions, at first find the period for each of them separately. Then find the minimum number which would get into itself for the whole amount of both periods. For example, function at =tgx*cos5x is given. For a tangent the period P, for a cosine of the 5th – the period 2P/5. The minimum number in which it is possible to find room for both of these periods, it 2P, thus, the required period – 2P.
5. If you find it difficult to act in an offered way or doubt the answer, try to act by definition. Take as function T period, it is more than zero. Substitute in the equation instead of x expression (x + T) and solve the received equality as though in T was the parameter or number. As a result you will find value of trigonometrical function and will be able to pick up the minimum period. For example, as a result of simplification at you the identity of sin (T/2)=0 turned out. The minimum value T at which it is carried out is equal 2P, it also will be the answer of a task.