Algebra and Trigonometry

# Key Equations

 Pythagorean identities $cos 2 θ+ sin 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ cos 2 θ+ sin 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ$ Even-odd identities $tan(−θ) = −tan θ cot(−θ) = −cot θ sin(−θ) = −sin θ csc(−θ) = −csc θ cos(−θ) = cos θ sec(−θ) = sec θ tan(−θ) = −tan θ cot(−θ) = −cot θ sin(−θ) = −sin θ csc(−θ) = −csc θ cos(−θ) = cos θ sec(−θ) = sec θ$ Reciprocal identities $sin θ = 1 csc θ cos θ = 1 sec θ tan θ = 1 cot θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ sin θ = 1 csc θ cos θ = 1 sec θ tan θ = 1 cot θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ$ Quotient identities $tan θ = sin θ cos θ cot θ = cos θ sin θ tan θ = sin θ cos θ cot θ = cos θ sin θ$
 Sum Formula for Cosine $cos( α+β )=cos α cos β−sin αsin β cos( α+β )=cos α cos β−sin αsin β$ Difference Formula for Cosine $cos( α−β )=cos α cos β+sin α sin β cos( α−β )=cos α cos β+sin α sin β$ Sum Formula for Sine $sin( α+β )=sin α cos β+cos α sin β sin( α+β )=sin α cos β+cos α sin β$ Difference Formula for Sine $sin( α−β )=sin α cos β−cos α sin β sin( α−β )=sin α cos β−cos α sin β$ Sum Formula for Tangent $tan( α+β )= tan α+tan β 1−tan α tan β tan( α+β )= tan α+tan β 1−tan α tan β$ Difference Formula for Tangent $tan( α−β )= tan α−tan β 1+tan α tan β tan( α−β )= tan α−tan β 1+tan α tan β$ Cofunction identities $sin θ = cos( π 2 −θ ) cos θ = sin( π 2 −θ ) tan θ = cot( π 2 −θ ) cot θ = tan( π 2 −θ ) sec θ = csc( π 2 −θ ) csc θ = sec( π 2 −θ ) sin θ = cos( π 2 −θ ) cos θ = sin( π 2 −θ ) tan θ = cot( π 2 −θ ) cot θ = tan( π 2 −θ ) sec θ = csc( π 2 −θ ) csc θ = sec( π 2 −θ )$
 Double-angle formulas $sin(2θ) = 2sin θ cos θ cos(2θ) = cos 2 θ− sin 2 θ = 1−2 sin 2 θ = 2 cos 2 θ−1 tan(2θ) = 2tan θ 1− tan 2 θ sin(2θ) = 2sin θ cos θ cos(2θ) = cos 2 θ− sin 2 θ = 1−2 sin 2 θ = 2 cos 2 θ−1 tan(2θ) = 2tan θ 1− tan 2 θ$ Reduction formulas $sin 2 θ = 1−cos(2θ) 2 cos 2 θ = 1+cos(2θ) 2 tan 2 θ = 1−cos(2θ) 1+cos(2θ) sin 2 θ = 1−cos(2θ) 2 cos 2 θ = 1+cos(2θ) 2 tan 2 θ = 1−cos(2θ) 1+cos(2θ)$ Half-angle formulas $sin α 2 = ± 1−cos α 2 cos α 2 = ± 1+cos α 2 tan α 2 = ± 1−cos α 1+cos α = sin α 1+cos α = 1−cos α sin α sin α 2 = ± 1−cos α 2 cos α 2 = ± 1+cos α 2 tan α 2 = ± 1−cos α 1+cos α = sin α 1+cos α = 1−cos α sin α$
 Product-to-sum Formulas $cos α cos β = 1 2 [cos(α−β)+cos(α+β)] sin α cos β = 1 2 [sin(α+β)+sin(α−β)] sin α sin β = 1 2 [cos(α−β)−cos(α+β)] cos α sin β = 1 2 [sin(α+β)−sin(α−β)] cos α cos β = 1 2 [cos(α−β)+cos(α+β)] sin α cos β = 1 2 [sin(α+β)+sin(α−β)] sin α sin β = 1 2 [cos(α−β)−cos(α+β)] cos α sin β = 1 2 [sin(α+β)−sin(α−β)]$ Sum-to-product Formulas $sin α+sin β = 2 sin( α+β 2 )cos( α−β 2 ) sin α−sin β = 2 sin( α−β 2 )cos( α+β 2 ) cos α−cos β = −2 sin( α+β 2 )sin( α−β 2 ) cos α+cos β = 2 cos( α+β 2 )cos( α−β 2 ) sin α+sin β = 2 sin( α+β 2 )cos( α−β 2 ) sin α−sin β = 2 sin( α−β 2 )cos( α+β 2 ) cos α−cos β = −2 sin( α+β 2 )sin( α−β 2 ) cos α+cos β = 2 cos( α+β 2 )cos( α−β 2 )$