Periodicity of Trigonometric Functions

Since `P_t` and `P_(t+360^0)` is same point on unit circle, then sines of corresponding angles are equal. Same can be said about cosines.

Therefore, `color(red)(sin(x+360^0)=sin(x))`, `color(blue)(cos(x+360^0)=cos(x))`.

In general, following equalities hold: `sin(x+360^0k)=sin(x)`, `cos(x+360^0k)=cos(x)`, where k is any integer number.

If argument x is expressed in radians, then `color(green)(sin(x+2pik)=sin(x)),color(205,92,92)(cos(x+2pik)=cos(x)),\ k in Z`.

For functions `y=tan(x)` and `y=cot(x)` following equalities hold: `color(135,10,80)(tan(x+pik)=tan(x)),color(139,0,0)(cot(x+pik)=cot(x)),\ k in Z`.

Therefore, any number of the form `2pik` is period of the functions `sin(x)` and `cos(x)`; and any number of the form `pik` is period of the functions `tan(x)` and `cot(x)`.

So, `2pi` is main period of `sin(x)` and `cos(x)`; and `pi` is main period of `tan(x)` and `cot(x)`.

Using even/odd property and property of periodicity, we can reduce trigonometric function of any angle reduce to trigonometric function of angle that belongs to the interval `[0^0,180^0]`.

Example. Find `sin(945^0)`.

We have that `sin(945^0)=sin(225^0+720^0)=sin(225^0+360^0*2)=sin(225^0)=sin(225^0-360^0)=`

`=sin(-135^0)=-sin(135^0)` .

Next, `sin(135^0)=sin(180^0-45^0)=sin(45^0)`.

But `sin(45^0)=(sqrt(2))/2`, therefore, `sin(945^0)=-sin(135^0)=-sin(45^0)=-(sqrt(2))/2`.