 Precalculus

# Key Equations

PrecalculusKey Equations

### Key Equations

 Pythagorean Identities $sin 2 θ+ cos 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ sin 2 θ+ cos 2 θ=1 1+ cot 2 θ= csc 2 θ 1+ tan 2 θ= sec 2 θ$ 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 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
 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 $sinα+sinβ=2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ=2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ=−2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ=2cos( α+β 2 )cos( α−β 2 ) sinα+sinβ=2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ=2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ=−2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ=2cos( α+β 2 )cos( α−β 2 )$
 Standard form of sinusoidal equation $y=Asin( Bt−C )+Dory=Acos( Bt−C )+D y=Asin( Bt−C )+Dory=Acos( Bt−C )+D$ Simple harmonic motion Damped harmonic motion $f( t )=a e −c t sin(ωt)orf( t )=a e −ct cos( ωt ) f( t )=a e −c t sin(ωt)orf( t )=a e −ct cos( ωt )$
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