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7.8.2: Key Equations

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    116137
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    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 )=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 )
    Standard form of sinusoidal equation y=Asin( BtC )+Dory=Acos( BtC )+D y=Asin( BtC )+Dory=Acos( BtC )+D
    Simple harmonic motion d=acos( ωt ) or d=asin( ωt ) d=acos( ωt ) or d=asin( ωt )
    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 )

    7.8.2: Key Equations is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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