# 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.