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Geometry Spot/Formulas/Induction

Induction (Reduction) Formulas

Express any angle as ±α + k·π/2. Odd k swaps function names; even k keeps them. Sign depends on quadrant.

Quick Reference
Core Transformations
Core Transformations (sin and cos)
Frequently used reductions
Sine Formulas
sin(−alpha)=−sinalpha\\sin(-\\alpha) = -\\sin\\alpha
sin(alpha+fracpi2)=cosalpha\\sin(\\alpha + \\frac{\\pi}{2}) = \\cos\\alpha
sin(fracpi2−alpha)=cosalpha\\sin(\\frac{\\pi}{2} - \\alpha) = \\cos\\alpha
sin(alpha+pi)=−sinalpha\\sin(\\alpha + \\pi) = -\\sin\\alpha
sin(pi−alpha)=sinalpha\\sin(\\pi - \\alpha) = \\sin\\alpha
Cosine Formulas
cos(−alpha)=cosalpha\\cos(-\\alpha) = \\cos\\alpha
cos(alpha+fracpi2)=−sinalpha\\cos(\\alpha + \\frac{\\pi}{2}) = -\\sin\\alpha
cos(fracpi2−alpha)=sinalpha\\cos(\\frac{\\pi}{2} - \\alpha) = \\sin\\alpha
cos(alpha+pi)=−cosalpha\\cos(\\alpha + \\pi) = -\\cos\\alpha
cos(pi−alpha)=−cosalpha\\cos(\\pi - \\alpha) = -\\cos\\alpha
Technique: Odd/Even and Quadrant
How to decide quickly
  1. Rewrite the angle as ±α + k·π/2.
  2. Check parity of k: odd → swap names; even → keep.
  3. Determine the quadrant of the original angle and apply the sign of the original function in that quadrant.