trigonometria

1251

ANEXO III
RELAÇÕES TRIGONOMÉTRICAS.
Sendo dado o triângulo abaixo.

a b c

As seguintes relações são válidas:
1) sen β =

b a c a a
5) sec β = c 3) tg β =

8) sen2 β + cos 2 β = 1

9) sec2 β − 1 = tg 2 β

2) cos β =

a b sen β
7) tg β = cos β
4) cosec β =

10) sen( α ± β ) = sen α cos β ± cos α sen β

b c 6) cotg β =

c b 11) cos( α ± β ) = cos α cos β m sen α sen β

1
1
12) sen α ± sen β = 2 ⋅ sen ( α ± β ) cos (α m β )
2
2
1
1
13) cos α + cos β = 2 ⋅ cos ( α + β ) cos (α − β )
2
2
1
1
14) cos α − cos β = − 2 ⋅ sen ( α + β ) sen (α − β )
2
2
1
15) sen α sen β = ⋅ cos ( α − β ) − cos (α + β )
2
1
16) cos α cos β = ⋅ cos ( α − β ) + cos ( α + β )
2
1
17) sen α cos β = ⋅ sen ( α − β ) + sen ( α + β )
2

[

]

[

]

[

]

18) sen 2 α = 2 sen α cos α
19) cos 2 α =cos 2 α − sen 2 α = 2 cos 2 α − 1 = 1 − 2sen 2 α
20) sen2

1
1
α = (1 − cos α )
2
2

21) cos 2

1
1
α = (1 + cos α )
2
2

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1

RELAÇÕES LOGARÍTIMICAS E EXPONENCIAIS.
1) e x ⋅ e y = e( x+ y )

2) (e x ) y = e x⋅ y = (e y ) x

3) log x ⋅ y = log x + log y

4) log

5) log x a = a ⋅ log x

6) ln x = (ln 10) ⋅ log x = 2,3026 ⋅ log x

x
= log x - log y y 7) log x = log e ln x = 0, 43429 ⋅ ln x

NÚMEROS COMPLEXOS.
Definimos j2 = − 1

j = −1

ou

e ± jθθ = cos θ ± j sen θ cos θ =

1 jθθ
(e + e − j θθ ),
2

sen θ =

1 jθθ
(e − e − jθθ )
2j

SÉRIE DE TAYLOR. f(x) = f(x 0 ) +

(x − x 0 ) ∂ f
(x − x 0 ) 2 ∂ 2 f
(
)
+
(
)
+...
1!
∂ x x= x
2!
∂ x 2 x =x
0

0

....+

(x − x 0 ) n ∂ n f
(
) n! ∂ x n x= x

SÉRIES COMUMENTE USADAS. x3 x5 x 7
+
−
+...
3! 5! 7! x 2 x4 x 6 cos x = 1 −
+
−
+ ...
2! 4! 6!
1
2
17 7 tg x = x + x 3 + x 5 + x +...
3
15
315
a 2 x 2 a 3x 3 ax e = 1 + ax +
+
+ ...
2!
3! x2 x3 ln(1+ x) = x +
−...
2
3
n(n - 1) x 2 n(n - 1) ⋅ (n - 2) x 3
(1 + x) n = 1+ nx +
+
+...
2!
3!

1) sen x = x