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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)`.