Formulário de Trigonometria

UNIVERSIDADE da MADEIRA
Departamento de Matemática e Engenharias
MATEMÁTICA I
Licenciaturas em Bioquı́mica e Quı́mica
1o Semestre 2005/2006
Fórmulas trigonométricas
Funções circulares
Funções hiperbólicas
cos (a + b) = cos a cos b − sin a sin b
cos (a − b) = cos a cos b + sin a sin b
sin (a + b) = sin a cos b + sin b cos a
sin (a − b) = sin a cos b − sin b cos a
tg a+tg b
tg (a + b) = 1−tg
a tg b
tg a−tg b
tg (a − b) = 1+tg
a tg b
ch (a + b) = ch a ch b + sh a sh b
ch (a − b) = ch a ch b − sh a sh b
sh (a + b) = sh a ch b + sh b ch a
sh (a + b) = sh a ch b − sh b ch a
th a+th b
th (a + b) = 1+th
a th b
th a−th b
th (a − b) = 1−th
a th b
cos 2a = cos2 a − sin2 a = 2 cos2 a − 1 =
2a
= 1 − 2 sin2 a = 1−tg
1+tg2 a
2tg a
sin 2a = 2 sin a cos a = 1+tg
2a
2tg a
tg 2a = 1−tg2 a
ch 2a = ch2 a + sh2 a = 2ch2 a − 1 =
1+th2 a
= 2sh2 a + 1 = 1−th
2
a
2th a
sh 2a = 2sh a ch a = 1−th2 a
2th a
th 2a = 1+th
2
a
a−b
cos a + cos b = 2 cos a+b
2 cos 2
a−b
cos a − cos b = 2 sin a+b
2 sin 2
a−b
sin a + sin b = 2 sin a+b
2 cos 2
a−b
sin a − sin b = 2 sin 2 cos a+b
2
sin(a+b)
tg a + tg b = cos
a cos b
sin(a−b)
tg a − tg b = cos
a cos b
a−b
ch a + ch b = 2ch a+b
2 ch 2
a−b
ch a − ch b = 2sh a+b
2 sh 2
a−b
sh a + sh b = 2sh a+b
2 ch 2
a−b
sh a − sh b = 2sh 2 ch a+b
2
th a + th b = sh(a+b)
ch a ch b
th a − th b = sh(a−b)
ch a ch b
cos a cos b = 21 [cos (a + b) + cos (a − b)]
sin a sin b = 21 [cos (a − b) − cos (a + b)]
sin a cos b = 21 [sin (a + b) + sin (a − b)]
ch a ch b =
sh a sh b =
sh a ch b =
cos x + i sin x = eix ; cos x − i sin x = e−ix
(cos x + i sin x)α = cos αx + i sin αx
(cos x − i sin x)α = cos αx − i sin αx
1
1
2
cos2 x = 1+tg
2 x ; sin x = 1+cotg2 x
ch x + sh x = ex ; ch x − sh x = e−x
(ch x + sh x)α = ch αx + sh αx
(ch x − sh x)α = ch αx − sh αx
th2 x
1
; sh2 x = 1−th
ch2 x = 1−th
2
2
x
x
1
2
1
2
1
2
[ch (a + b) + ch (a − b)]
[ch (a + b) − ch (a − b)]
[sh (a + b) + sh (a − b)]