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 $sinα+sinβ = 2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ = 2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ = −2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ = 2cos( α+β 2 )cos( α−β 2 ) sinα+sinβ = 2sin( α+β 2 )cos( α−β 2 ) sinα−sinβ = 2sin( α−β 2 )cos( α+β 2 ) cosα−cosβ = −2sin( α+β 2 )sin( α−β 2 ) cosα+cosβ = 2cos( α+β 2 )cos( α−β 2 )$

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

• If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
• If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution: