Induction (Reduction) Formulas

Turn any angle of the form ±α + k·π/2 into a function of an acute angle α. Two rules guide the process: (1) Odd change, even no change — when k is odd, the function name changes (sin ↔ cos, tan ↔ cot); when k is even, the function name stays. (2) Sign by quadrant — determine the quadrant of the original angle and use the sign of the original function in that quadrant.

Framework: ±α + k·π/2

Universal reduction to functions of an acute angle α

General form: any angle can be written as ±α + k·π/2 with integer k.
Odd change, even no change: if k is odd, sin ↔ cos (and tan ↔ cot). If k is even, the function name remains.
Sign by quadrant: treat α as acute for quadrant analysis only, locate the original angle, and apply the sign of the original function in that quadrant.

Worked Examples

Common patterns for sine and cosine

Angle: −α

sin(−α) = −sin α

cos(−α) = cos α

Angle: α + π/2

sin(α + π/2) = cos α

cos(α + π/2) = −sin α

Angle: π/2 − α

sin(π/2 − α) = cos α

cos(π/2 − α) = sin α

Angle: α + π

sin(α + π) = −sin α

cos(α + π) = −cos α

Angle: π − α

sin(π − α) = sin α

cos(π − α) = −cos α

Angle: α + 3π/2

sin(α + 3π/2) = −cos α

cos(α + 3π/2) = sin α

Angle: 3π/2 − α

sin(3π/2 − α) = −cos α

cos(3π/2 − α) = −sin α

Angle: α + 2π

sin(α + 2π) = sin α

cos(α + 2π) = cos α

Technique: Standardize the Angle

Convert to ±α + k·π/2 before applying rules

For example, sin(5π/2 + α) → α + 5·π/2 (k = 5 is odd). Name changes: sin → cos. Quadrant: equivalent to π/2 + α (since +2π cycles), which is in QII where sine is positive. Result: sin(5π/2 + α) = cos α.