Relações Trigonométricas

Relações básicas

sen2 α + cos2 α = 1

tan α  cot α = 1

1 + tan2 α = 1 / cos2 α

1 + cot2 α = 1 / sen2 α

Relações com quadrantes

 
Obs: valores de ângulos em graus. Conversão para radianos:

90 → π/2     180 → π     270 → 3π/2     360 → 2π

sen (90 + α) = + cos α

sen (90 − α) = + cos α

sen (180 + α) = − sen α

sen (180 − α) = + sen α

cos (90 + α) = − sen α

cos (90 − α) = + sen α

cos (180 + α) = − cos α

cos (180 − α) = − cos α

tab (90 + α) = − cot α

tan (90 − α) = + cot α

tan (180 + α) = + tan α

tan (180 − α) = − tan α

cot (90 + α) = − tan α

cot (90 − α) = + tan α

cot (180 + α) = + cot α

cot (180 − α) = − cot α

sen (270 + α) = − cos α

sen (270 − α) = − cos α

sen (360 + α) = + sen α

sen (360 − α) = − sen α

cos (270 + α) = + sen α

cos (270 − α) = − sen α

cos (360 + α) = + cos α

cos (360 − α) = + cos α

tan (270 + α) = − cot α

tan (270 − α) = + cot α

tan (360 + α) = + tan α

tan (360 − α) = − tan α

cot (270 + α) = − tan α

cot (270 − α) = + tan α

cot (360 + α) = + cot α

cot (360 − α) = − cot α

sen (−α) = − sen α

cos (−α) = + cos α

tan (−α) = − tan α

cot (−α) = − cot α

sen (α ± k 360) = + sen α

cos (α ± k 360) = + cos α

tan (α ± k 180) = + tan α

cot (α ± k 180) = + cot α

O símbolo k significa um número inteiro e positivo.

Relações com soma / diferença de ângulos.

 
sen (α [pic 1] β) = sen α cos β [pic 2] cos α sen β

cos (α [pic 3] β) = cos α cos β [pic 4] sen α sen β

tan (α [pic 5] β) = (tan α [pic 6] tan β) / (1 [pic 7] tan α tan β)

cot (α [pic 8] β) = (cot α cot β [pic 9] 1) / (cot β [pic 10] cot α)

Relações com soma / diferença / produto de funções

sen α + sen β = 2 sen (α + β)/2 . cos (α − β)/2

sen α − sen β = 2 cos (α + β)/2 . sen (α − β)/2

cos α + cos β = 2 cos (α + β)/2 . cos (α − β)/2

cos α − cos β = − 2 sen (α + β)/2 . sen (α − β)/2

a sen x + b cos x = √ (a2 + b2) sen (x + φ)

onde φ = arctan b/a se a ≥ 0 ou φ = arctan b/a ± π se a < 0

tan α [pic 11] tan β = sen (α [pic 12] β) / (cos α cos β)

cot α [pic 13] cot β = sen (β [pic 14] α) / (sen α sen β)

sen α sen β = (1/2) cos (α − β) − (1/2) cos (α + β)

sen α cos β = (1/2) sen (α + β) + (1/2) sen (α − β)

cos α cos β = (1/2) cos (α + β) + (1/2) cos (α − β)

tan α tan β = (tan α + tan β) / (cot α + cot β) = − (tan α − tan β) / (cot α − cot β)

cot α cot β = (cot α + cot β) / (tan α + tan β) = − (cot α − cot β) / (tan α − tan β)

cot α tan β = (cot α + tan β) / (tan α + cot β) = − (cot α − tan β) / (tan α − cot β)

Relações diversas

sen α = 2 sen α/2 . cos α/2

cos α = cos2 α/2 − sen2 α/2

tan α = sen α / cos α

cot α = cos α / sen α

sen α = tan α / √(1 + tan2 α)

cos α = cot α / √(1 + cot2 α)

tan α = sen α / √(1 − sen2 α)

cot α = cos α / √(1 − cos2 α)

sen α = √(cos2 α − cos 2α)

cos α = 1 − 2 sen2 α/2

tan α = √[ (1/cos2 α) − 1 ]

cot α = √[ (1/sen2 α) − 1 ]

sen α = √[ (1 − cos 2α) / 2 ]

cos α = √[ (1 + cos 2α) / 2 ]

tan α = [ √(1 − cos2 α) ] / cos α

cot α = [ √(1 − sen2 α) ] / sen α

sen α = 1 / √(1 + cot2 α)

cos α = 1 / √(1 + tan2 α)

sen 2α = 2 sen α cos α

cos 2α = cos2 α − sen2 α

cos 2α = 2 cos2 α − 1

cos 2α = 1 − 2 sen2 α

tan 2α = 2 tan α / (1 − tan2 α)

tan 2α = 2 / (cot α − tan α)

cot 2α = (cot2 α − 1) / (2 cot α)

cot 2α = (1/2) cot α − (1/2) tan α

sen α/2 = √[ (1 − cos α) / 2 ]

cos α/2 = √[ (1 + cos α) / 2 ]