Relações Trigonométricas
Por: Alan Gomes • 11/10/2016 • Exam • 1.291 Palavras (6 Páginas) • 257 Visualizações
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 ]
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