# Reduction Formulas

Reduction formulas are formulas that allow to reduce function with argument of the form (pi n)/2+-alpha,n in ZZ to function with argument alpha.

Example. Find sin(pi/2+alpha).

We have that sin(pi/2+alpha)=sin(pi/2)cos(alpha)+cos(pi/2)sin(alpha)=1*cos(alpha)+0*sin(alpha)=cos(alpha).

In a similar manner we can obtain other reduction formulas, that are given in following table:

 Function Argument t pi/2-alpha pi/2+alpha pi-alpha pi+alpha (3pi)/2-alpha (3pi)/2+alpha 2pi-alpha sin(t) cos(alpha) cos(alpha) sin(alpha) -sin(alpha) -cos(alpha) -cos(alpha) -sin(alpha) cos(t) sin(alpha) -sin(alpha) -cos(alpha) -cos(alpha) -sin(alpha) sin(alpha) cos(alpha) tan(t) cot(alpha) -cot(alpha) -tan(alpha) tan(alpha) cot(alpha) -cot(alpha) -tan(alpha) cot(t) tan(alpha) -tan(alpha) -cot(alpha) cot(alpha) tan(alpha) -tan(alpha) -cot(alpha)

For example, this table tells us that cos(pi+alpha)=-cos(alpha) and tan((3pi)/2-alpha)=cot(alpha).

To make easy remembering of reduction formulas, following rules are used:

1. In the right side of formula we write sign which has expression in left side, provided 0<alpha<pi/2. For example, pi/2<pi/2+alpha<pi. Cosine of such argument is negative, so we write "-" in the right side (cos(pi/2+alpha)=-sin(alpha)).
2. If in left side of formulas angle equals pi/2+-alpha or (3pi)/2+-alpha then we write cos instead of sin,tan instead of cot; and vice versa. For example, cot((3pi)/2+alpha)=-tan(alpha). If angle equals pi+-alpha or 2pi-alpha then we don't change functions. For example, cos(pi+alpha)=-cos(alpha).