Precalculus

# A | Basic Functions and Identities

PrecalculusA | Basic Functions and Identities

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## Trigonometric Identities

 Pythagorean Identities $cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ cos 2 θ+ sin 2 θ=1 1+ tan 2 θ= sec 2 θ 1+ cot 2 θ= csc 2 θ$ Even-Odd Identities $cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=−sinθ tan(−θ)=−tanθ csc(−θ)=−cscθ cot(−θ)=−cotθ cos(−θ)=cosθ sec(−θ)=secθ sin(−θ)=−sinθ tan(−θ)=−tanθ csc(−θ)=−cscθ cot(−θ)=−cotθ$ Cofunction Identities $cosθ=sin( π 2 −θ ) sinθ=cos( π 2 −θ ) tanθ=cot( π 2 −θ ) cotθ=tan( π 2 −θ ) secθ=csc( π 2 −θ ) cscθ=sec( π 2 −θ ) cosθ=sin( π 2 −θ ) sinθ=cos( π 2 −θ ) tanθ=cot( π 2 −θ ) cotθ=tan( π 2 −θ ) secθ=csc( π 2 −θ ) cscθ=sec( π 2 −θ )$ Fundamental Identities $tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ tanθ= sinθ cosθ secθ= 1 cosθ cscθ= 1 sinθ cotθ= 1 tanθ = cosθ sinθ$ Sum and Difference Identities $cos(α+β)=cosαcosβ−sinαsinβ cos(α−β)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(α−β)=sinαcosβ−cosαsinβ tan(α+β)= tanα+tanβ 1−tanαtanβ tan(α−β)= tanα−tanβ 1+tanαtanβ cos(α+β)=cosαcosβ−sinαsinβ cos(α−β)=cosαcosβ+sinαsinβ sin(α+β)=sinαcosβ+cosαsinβ sin(α−β)=sinαcosβ−cosαsinβ tan(α+β)= tanα+tanβ 1−tanαtanβ tan(α−β)= tanα−tanβ 1+tanαtanβ$ Double-Angle Formulas $sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ− sin 2 θ cos(2θ)=1−2 sin 2 θ cos(2θ)=2 cos 2 θ−1 tan(2θ)= 2tanθ 1− tan 2 θ sin(2θ)=2sinθcosθ cos(2θ)= cos 2 θ− sin 2 θ cos(2θ)=1−2 sin 2 θ cos(2θ)=2 cos 2 θ−1 tan(2θ)= 2tanθ 1− tan 2 θ$ Half-Angle Formulas $sin α 2 =± 1−cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1−cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1−cosα sinα sin α 2 =± 1−cosα 2 cos α 2 =± 1+cosα 2 tan α 2 =± 1−cosα 1+cosα tan α 2 = sinα 1+cosα tan α 2 = 1−cosα sinα$ Reduction Formulas $sin 2 θ= 1−cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1−cos( 2θ ) 1+cos( 2θ ) sin 2 θ= 1−cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1−cos( 2θ ) 1+cos( 2θ )$ 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 $sinα+sinβ=2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ=2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ=−2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ=2cos( α+β 2 )cos( α−β 2 ) sinα+sinβ=2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ=2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ=−2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ=2cos( α+β 2 )cos( α−β 2 )$ Law of Sines $sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ sinα a = sinβ b = sinγ c a sinα = b sinβ = c sinγ$ Law of Cosines $a 2 = b 2 + c 2 −2bccosα b 2 = a 2 + c 2 −2accosβ c 2 = a 2 + b 2 −2abcosγ a 2 = b 2 + c 2 −2bccosα b 2 = a 2 + c 2 −2accosβ c 2 = a 2 + b 2 −2abcosγ$
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