Algebra and Trigonometry 2e

# Key Equations

### 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 $cos( α+β )=cosαcosβ−sinαsinβ cos( α+β )=cosαcosβ−sinαsinβ$ Difference Formula for Cosine $cos( α−β )=cosαcosβ+sinαsinβ cos( α−β )=cosαcosβ+sinαsinβ$ Sum Formula for Sine $sin( α+β )=sinαcosβ+cosαsinβ sin( α+β )=sinαcosβ+cosαsinβ$ Difference Formula for Sine $sin( α−β )=sinαcosβ−cosαsinβ sin( α−β )=sinαcosβ−cosαsinβ$ Sum Formula for Tangent $tan( α+β )= tanα+tanβ 1−tanαtanβ tan( α+β )= tanα+tanβ 1−tanαtanβ$ Difference Formula for Tangent $tan( α−β )= tanα−tanβ 1+tanαtanβ tan( α−β )= tanα−tanβ 1+tanαtanβ$ 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 $sin(2θ) = 2sinθcosθ cos(2θ) = cos 2 θ− sin 2 θ = 1−2 sin 2 θ = 2 cos 2 θ−1 tan(2θ) = 2tanθ 1− tan 2 θ sin(2θ) = 2sinθcosθ cos(2θ) = cos 2 θ− sin 2 θ = 1−2 sin 2 θ = 2 cos 2 θ−1 tan(2θ) = 2tanθ 1− tan 2 θ$ 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 $sin α 2 = ± 1−cosα 2 cos α 2 = ± 1+cosα 2 tan α 2 = ± 1−cosα 1+cosα = sinα 1+cosα = 1−cosα sinα sin α 2 = ± 1−cosα 2 cos α 2 = ± 1+cosα 2 tan α 2 = ± 1−cosα 1+cosα = sinα 1+cosα = 1−cosα sinα$
 Product-to-sum Formulas $cosαcosβ = 1 2 [cos(α−β)+cos(α+β)] sinαcosβ = 1 2 [sin(α+β)+sin(α−β)] sinαsinβ = 1 2 [cos(α−β)−cos(α+β)] cosαsinβ = 1 2 [sin(α+β)−sin(α−β)] cosαcosβ = 1 2 [cos(α−β)+cos(α+β)] sinαcosβ = 1 2 [sin(α+β)+sin(α−β)] sinαsinβ = 1 2 [cos(α−β)−cos(α+β)] cosαsinβ = 1 2 [sin(α+β)−sin(α−β)]$ Sum-to-product Formulas $sinα+sinβ = 2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ = 2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ = −2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ = 2cos( α+β 2 )cos( α−β 2 ) sinα+sinβ = 2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ = 2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ = −2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ = 2cos( α+β 2 )cos( α−β 2 )$
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