Relações Trigonométricas

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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 ]

tan α/2 = sen α / (1 + cos α)

cot α/2 = sen α / (1 − cos α)

tan α/2 = (1 − cos α) / sen α

cot α/2 = (1 + cos α) / sen α

tan α/2 = √[ (1 − cos α) / (1 + cos α) ]

cot α/2 = √[ (1 + cos α) / (1 − cos α) ]

Relações de funções inversas

Relações básicasarccos x = π/2 − arcsen x

arccot x = π/2 − arctan x

Relações de soma / subtraçãoarcsen a [pic 15] arcsen b = arcsen [ a √(1 − b2) [pic 16] b √(1 − a2) ]

arccos a [pic 17] arccos b = arccos [ a b [pic 18] √(1 − a2) √(1 − b2) ]

arctan a [pic 19] arctan b = arctan [ (a [pic 20] b) / (1 [pic 21] a b) ]

arccot a [pic 22] arccot b = arccot [ (a b [pic 23] 1) / (b [pic 24] a) ]

Relações entre funções (para x > 0)

arcsen x = arccos √(1 − x2) = arctan [ x / √(1 − x2) ] = arccot [ √(1 − x2) / x ]

arccso x = arcsen √(1 − x2) = arctan [ √(1 − x2) / x ] = arccot [ x / √(1 − x2) ]

arctan x = arcsen [ x / √(1 + x2) ] = arccos [ 1 / √(1 + x2) ] = arccot [ 1 / x ]

arccot x = arcsen [ 1 / √(1 + x2) ] = arccos [ x / √(1 + x2) ] = arctan [ 1 / x ]

Para x , valem as relações:

arcsen (−x) = − arcsen x

arccos (−x) = π − arccos x

arctan (−x) = − arctan x

arccot (−x) = π − arccot x