Double-angle and reduction formulas

Reduction formulas reduce the functions of any angle to functions of an acute angle. Sine and cosine have period 2π: sin(α + 2πk) = sin α, cos(α + 2πk) = cos α (k ∈ Z); tangent and cotangent have period π: tan(α + πk) = tan α. A function is called periodic if there is a number T ≠ 0 with f(x − T) = f(x) = f(x + T); the smallest positive T is the main period. Also: sin(π − α) = sin α, cos(π − α) = −cos α; sin(π/2 − α) = cos α, cos(π/2 − α) = sin α. Procedure: 1) remove whole turns (whole multiples of 2π or 360°); 2) write the remaining angle as π ± α or π/2 ± α; 3) apply the formula and decide the sign from the quadrant.