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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) = tanθ cot(θ) = cotθ sin(θ) = sinθ csc(θ) = cscθ cos(θ) = cosθ sec(θ) = secθ tan(θ) = tanθ cot(θ) = cotθ sin(θ) = sinθ csc(θ) = cscθ cos(θ) = cosθ sec(θ) = secθ
Reciprocal identities sinθ = 1 cscθ cosθ = 1 secθ tanθ = 1 cotθ cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ sinθ = 1 cscθ cosθ = 1 secθ tanθ = 1 cotθ cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ
Quotient identities tanθ = sinθ cosθ cotθ = cosθ sinθ tanθ = sinθ cosθ cotθ = cosθ sinθ
Sum Formula for Cosine cos( α+β )=cosαcosβsinαsinβ cos( α+β )=cosαcosβsinαsinβ
Difference Formula for Cosine cos( αβ )=cosαcosβ+sinαsinβ cos( αβ )=cosαcosβ+sinαsinβ
Sum Formula for Sine sin( α+β )=sinαcosβ+cosαsinβ sin( α+β )=sinαcosβ+cosαsinβ
Difference Formula for Sine sin( αβ )=sinαcosβcosαsinβ sin( αβ )=sinαcosβcosαsinβ
Sum Formula for Tangent tan( α+β )= tanα+tanβ 1tanαtanβ tan( α+β )= tanα+tanβ 1tanαtanβ
Difference Formula for Tangent tan( αβ )= tanαtanβ 1+tanαtanβ tan( αβ )= tanαtanβ 1+tanαtanβ
Cofunction identities sinθ = cos( π 2 θ ) cosθ = sin( π 2 θ ) tanθ = cot( π 2 θ ) cotθ = tan( π 2 θ ) secθ = csc( π 2 θ ) cscθ = sec( π 2 θ ) sinθ = cos( π 2 θ ) cosθ = sin( π 2 θ ) tanθ = cot( π 2 θ ) cotθ = tan( π 2 θ ) secθ = csc( π 2 θ ) cscθ = sec( π 2 θ )
Double-angle formulas sin(2θ) = 2sinθcosθ cos(2θ) = cos 2 θ sin 2 θ = 12 sin 2 θ = 2 cos 2 θ1 tan(2θ) = 2tanθ 1 tan 2 θ sin(2θ) = 2sinθcosθ cos(2θ) = cos 2 θ sin 2 θ = 12 sin 2 θ = 2 cos 2 θ1 tan(2θ) = 2tanθ 1 tan 2 θ
Reduction formulas sin 2 θ = 1cos(2θ) 2 cos 2 θ = 1+cos(2θ) 2 tan 2 θ = 1cos(2θ) 1+cos(2θ) sin 2 θ = 1cos(2θ) 2 cos 2 θ = 1+cos(2θ) 2 tan 2 θ = 1cos(2θ) 1+cos(2θ)
Half-angle formulas sin α 2 = ± 1cosα 2 cos α 2 = ± 1+cosα 2 tan α 2 = ± 1cosα 1+cosα = sinα 1+cosα = 1cosα sinα sin α 2 = ± 1cosα 2 cos α 2 = ± 1+cosα 2 tan α 2 = ± 1cosα 1+cosα = sinα 1+cosα = 1cosα sinα
Product-to-sum Formulas cosαcosβ = 1 2 [cos(αβ)+cos(α+β)] sinαcosβ = 1 2 [sin(α+β)+sin(αβ)] sinαsinβ = 1 2 [cos(αβ)cos(α+β)] cosαsinβ = 1 2 [sin(α+β)sin(αβ)] cosαcosβ = 1 2 [cos(αβ)+cos(α+β)] sinαcosβ = 1 2 [sin(α+β)+sin(αβ)] sinαsinβ = 1 2 [cos(αβ)cos(α+β)] cosαsinβ = 1 2 [sin(α+β)sin(αβ)]
Sum-to-product Formulas sinα+sinβ = 2sin( α+β 2 )cos( αβ 2 ) sinαsinβ = 2sin( αβ 2 )cos( α+β 2 ) cosαcosβ = 2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ = 2cos( α+β 2 )cos( αβ 2 ) sinα+sinβ = 2sin( α+β 2 )cos( αβ 2 ) sinαsinβ = 2sin( αβ 2 )cos( α+β 2 ) cosαcosβ = 2sin( α+β 2 )sin( αβ 2 ) cosα+cosβ = 2cos( α+β 2 )cos( αβ 2 )
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