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Induction (Reduction) Formulas

Any angle can be expressed as ±α + k·π/2. Use two rules: (1) when k is odd, function names swap (sin ↔ cos, tan ↔ cot); when k is even, names stay. (2) Determine the sign by the quadrant of the original angle. Below is a concise reference.

Core Transformations (sin and cos)

Frequently used reductions

Sine

sin(−α) = −sin α
sin(α + π/2) = cos α
sin(π/2 − α) = cos α
sin(α + π) = −sin α
sin(π − α) = sin α
sin(α + 3π/2) = −cos α
sin(3π/2 − α) = −cos α
sin(α + 2π) = sin α

Cosine

cos(−α) = cos α
cos(α + π/2) = −sin α
cos(π/2 − α) = sin α
cos(α + π) = −cos α
cos(π − α) = −cos α
cos(α + 3π/2) = sin α
cos(3π/2 − α) = −sin α
cos(α + 2π) = cos α

Technique: Odd/Even and Quadrant

How to decide quickly

Rewrite the angle as ±α + k·π/2.
Check parity of k: odd → swap names; even → keep.
Determine the quadrant of the original angle and apply the sign of the original function in that quadrant.