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Transcript of gma00116-aula-18.pdf
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7/25/2019 gma00116-aula-18.pdf
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Pr-Clculo
Humberto Jos Bortolossi
Departamento de Matemtica Aplicada
Universidade Federal Fluminense
Aula 18
7 de julho de 2011
Aula 18 Pr-Clculo 1
http://find/http://goback/ -
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A funo tangente
Aula 18 Pr-Clculo 2
http://find/http://goback/ -
7/25/2019 gma00116-aula-18.pdf
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A funo tangente
f(x) =tg(x) =sen(x)
cos(x)
Qual o domnio natural da funo tangente?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 3
http://find/ -
7/25/2019 gma00116-aula-18.pdf
4/98
A funo tangente
f(x) =tg(x) =sen(x)
cos(x)
Qual o domnio natural da funo tangente?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 4
A f
http://find/http://goback/ -
7/25/2019 gma00116-aula-18.pdf
5/98
A funo tangente
f(x) =tg(x) =sen(x)cos(x)
Qual o domnio natural da funo tangente?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 5
A f t t
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo tangente
f(x) =tg(x) =sen(x)cos(x)
Qual o domnio natural da funo tangente?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 6
A f t t
http://find/http://goback/ -
7/25/2019 gma00116-aula-18.pdf
7/98
A funo tangente
f(x) =tg(x) =sen(x)cos(x)
Qual o domnio natural da funo tangente?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 7
O fi d f t t
http://find/http://goback/ -
7/25/2019 gma00116-aula-18.pdf
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O grfico da funo tangente
(http://www.uff.br/cdme/ftr/ftr -html/ftr-tangente-rad-br.htmlou http://www.cdme.im-uff.mat.br/ftr/ftr-html/ftr-tangente-rad-br.html)
Aula 18 Pr-Clculo 8
http://www.uff.br/cdme/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.uff.br/cdme/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.cdme.im-uff.mat.br/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.cdme.im-uff.mat.br/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.cdme.im-uff.mat.br/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.cdme.im-uff.mat.br/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://www.uff.br/cdme/ftr/ftr-html/ftr-tangente-rad-br.htmlhttp://find/ -
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A funo secante
Aula 18 Pr-Clculo 9
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
f(x) =sec(x) = 1cos(x)
Qual o domnio natural da funo secante?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 10
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
f(x) =sec(x) = 1cos(x)
Qual o domnio natural da funo secante?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 11
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
f(x) =sec(x) = 1cos(x)
Qual o domnio natural da funo secante?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 12
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
f(x) =sec(x) = 1cos(x)
Qual o domnio natural da funo secante?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 13
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
f(x) =sec(x) = 1cos(x)
Qual o domnio natural da funo secante?
D={x R| cos(x)=0}={x R|x=/2 + k , comk Z}
Aula 18 Pr-Clculo 14
A funo secante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo secante
Aula 18 Pr-Clculo 15
http://find/ -
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A funo cossecante
Aula 18 Pr-Clculo 16
A funo cossecante
http://find/http://goback/ -
7/25/2019 gma00116-aula-18.pdf
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A funo cossecante
f(x) =cossec(x) = 1
sen(x)
Qual o domnio natural da funo cossecante?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 17
A funo cossecante
http://find/http://goback/ -
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A funo cossecante
f(x) =cossec(x) = 1
sen(x)
Qual o domnio natural da funo cossecante?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 18
A funo cossecante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo cossecante
f(x) =cossec(x) = 1
sen(x)
Qual o domnio natural da funo cossecante?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 19
A funo cossecante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo cossecante
f(x) =cossec(x) = 1
sen(x)
Qual o domnio natural da funo cossecante?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 20
A funo cossecante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
21/98
A funo cossecante
f(x) =cossec(x) = 1
sen(x)
Qual o domnio natural da funo cossecante?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 21
A funo cossecante
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo cossecante
Aula 18 Pr-Clculo 22
http://find/ -
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A funo cotangente
Aula 18 Pr-Clculo 23
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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g
f(x) =cotg(x) = cos(x)
sen(x)
Qual o domnio natural da funo cotangente?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 24
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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g
f(x) =cotg(x) = cos(x)
sen(x)
Qual o domnio natural da funo cotangente?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 25
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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g
f(x) =cotg(x) = cos(x)
sen(x)
Qual o domnio natural da funo cotangente?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 26
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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g
f(x) =cotg(x) = cos(x)
sen(x)
Qual o domnio natural da funo cotangente?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 27
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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g
f(x) =cotg(x) = cos(x)
sen(x)
Qual o domnio natural da funo cotangente?
D={x R| sen(x)=0}={x R| x=k , comk Z}
Aula 18 Pr-Clculo 28
A funo cotangente
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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Aula 18 Pr-Clculo 29
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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A funo arco seno
Aula 18 Pr-Clculo 30
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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f: R Rx y=f(x) =sen(x) no inversvel, pois no injetiva.
Aula 18 Pr-Clculo 31
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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f: [/2,+/2] [1,+1]x y=f(x) =sen(x) inversvel, pois bijetiva.
Aula 18 Pr-Clculo 32
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
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f1 : [1,+1] [/2, +/2]x y=f1(x) =arcsen(x) sua funo inversa.
Aula 18 Pr-Clculo 33
Exemplo
http://find/ -
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f1 : [1,+1] [/2, +/2]x y=f1(x) =arcsen(x) sua funo inversa.
Aula 18 Pr-Clculo 34
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
35/98
Mostre que cos(arcsen
(x
)) =
1
x2, parax
(1,+
1)
.
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) = 1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 35
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
36/98
Mostre que cos(arcsen
(x
)) =
1
x2, parax
(1,+
1)
.
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) = 1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 36
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
37/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) = 1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 37
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
38/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) = 1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 38
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
39/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 39
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
40/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 40
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
41/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 41
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
42/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 42
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
43/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 43
A funo arco seno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
44/98
Mostre que cos(arcsen(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arcsen(x))]2 + [sen(arcsen(x))]2 =1 [cos(arcsen(x))]2 + x2 =1
[cos(arcsen(x))]2 =1 x2
[cos(arcsen(x))]2 =
1 x2
| cos(arcsen(x))|=
1 x2
cos(arcsen(x)) =
1
x2,
pois sex(1,+1), ento arcsen(x)(/2,+/2)e, assim, cos(arcsen(x))> 0.
Aula 18 Pr-Clculo 44
http://find/ -
7/25/2019 gma00116-aula-18.pdf
45/98
A funo arco cosseno
Aula 18 Pr-Clculo 45
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
46/98
f: R Rx y=f(x) =cos(x) no inversvel, pois no injetiva.
Aula 18 Pr-Clculo 46
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
47/98
f: [0, ] [1,+1]x y=f(x) =cos(x) inversvel, pois bijetiva.
Aula 18 Pr-Clculo 47
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
48/98
f1 : [1,+1] [0, ]x y=f1(x) =arccos(x) sua funo inversa.
Aula 18 Pr-Clculo 48
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
49/98
f1 : [1,+1] [0, ]x y=f1(x) =arccos(x) sua funo inversa.
Aula 18 Pr-Clculo 49
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
50/98
Mostre que sen(
arccos(x
)) =
1
x2, parax
(1,+
1)
.
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 50
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
51/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 51
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
52/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 52
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
53/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 53
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
54/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 54
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
55/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 55
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
56/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 56
A funo arco cosseno
http://find/ -
7/25/2019 gma00116-aula-18.pdf
57/98
Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 57
A funo arco cosseno
http://find/ -
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Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 58
A funo arco cosseno
http://find/ -
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Mostre que sen(arccos(x)) =
1x2, parax
(
1,+1).
Demonstrao.
[cos(arccos(x))]2 + [sen(arccos(x))]2 =1 x2 + [sen(arccos(x))]2 =1
[sen(arccos(x))]2
=1 x2
[sen(arccos(x))]2 =
1 x2
| sen(arccos(x))|=
1 x2
sen(arccos(x)) =
1 x2,
pois sex(1,+1), ento arccos(x)(0, )e, assim, sen(arcsen(x))> 0.
Aula 18 Pr-Clculo 59
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A funo arco tangente
Aula 18 Pr-Clculo 60
A funo arco tangente
f : R {/2 + k | k Z} R
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f: R {/2 + k | k Z} Rx y=f(x) =tg(x) no inversvel.
Aula 18 Pr-Clculo 61
A funo arco tangente
f : (/2,+/2) R i l i bij i
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f: ( /2,+/2) Rx y=f(x) =tg(x) inversvel, pois bijetiva.
Aula 18 Pr-Clculo 62
A funo arco tangente
f1 : R (/2,+/2) f i
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f : R ( /2,+/2)x y=f1(x) =arctg(x) sua funo inversa.
Aula 18 Pr-Clculo 63
A funo arco tangente
f1 : R (/2,+/2) f i
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( / ,+ / )
x y=f1(x) =arctg(x) sua funo inversa.
Aula 18 Pr-Clculo 64
A funo arco tangente
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 65
A funo arco tangente
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 66
A funo arco tangente
2 2
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 67
A funo arco tangente
2 2
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 68
A funo arco tangente
M 2( ( )) 1 2
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 69
A funo arco tangente
M t 2( t ( )) 1 2 R
http://find/ -
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 70
A funo arco tangente
M t 2( t ( )) 1 + 2 R
http://find/ -
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Mostre que sec2(arctg(x)) =1 + x2, parax R.
Demonstrao.
[cos(arctg(x))]2 + [sen(arctg(x))]2 =1
[cos(arctg(x))]2 + [sen(arctg(x))]2
cos2(arctg(x)) =
1
cos2(arctg(x))
1 + tg2(arctg(x)) =sec2(arctg(x))
1 + x2 =sec2(arctg(x))
sec2(arctg(x))= 1 + x2.
Aula 18 Pr-Clculo 71
http://find/ -
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As frmulas de adio
Aula 18 Pr-Clculo 72
As frmulas de adio
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 73
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 74
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 75
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 76
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 77
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 78
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 79
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 80
As frmulas de adio
http://goback/http://find/http://goforward/http://goforward/http://find/http://goback/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 81
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 82
As frmulas de adio
http://find/http://find/ -
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OA = cos( + ),
OE = cos(),
EC = sen(),
AB = DE = sen() sen(),OB = cos() cos().
cos( + ) =OA= OB AB= cos() cos() sen() sen().
Aula 18 Pr-Clculo 83
As frmulas de adio
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 84
As frmulas de adio
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 85
As frmulas de adio
http://find/ -
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 86
As frmulas de adio
http://find/ -
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 87
As frmulas de adio
http://find/ -
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 88
As frmulas de adio
http://find/http://goback/ -
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cos( + ) = cos() cos() sen() sen().
cos( ) = cos( + ())= cos()
cos(
)
sen()
sen(
)
= cos() cos() + sen() sen().
cos(2) =cos( + ) =cos2()
sen2().
Aula 18 Pr-Clculo 89
As frmulas de adio
J vimos que:t
(t)
t
(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr-Clculo 90
As frmulas de adio
J vimos que:
sen
+ t
cos(t) cos
+ t
sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr-Clculo 91
As frmulas de adio
J vimos que:
sen
+ t
cos(t) cos
+ t
sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr Clculo 92
As frmulas de adio
J vimos que:
sen
+ t
cos(t) cos
+ t
sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr Clculo 93
As frmulas de adio
J vimos que:
sen
+ t
= cos(t) cos
+ t
= sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr Clculo 94
As frmulas de adio
J vimos que:
sen
+ t
= cos(t) cos
+ t
= sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr Clculo 95
As frmulas de adio
J vimos que:
sen
+ t
= cos(t) cos
+ t
= sen(t)
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sen
2+ t
= cos(t), cos
2
+ t
= sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr-Clculo 96
As frmulas de adio
J vimos que:
sen
+ t
= cos(t) cos
+ t
= sen(t)
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sen
2+ t
cos(t), cos
2
+ t
sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr-Clculo 97
As frmulas de adio
J vimos que:
sen
+ t
= cos(t), cos
+ t
= sen(t).
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sen
2+ t
cos(t), cos
2
+ t
sen(t).
Agora:
sen( + ) = cos
2+ +
= cos
2 + cos() + sen
2 + sen()
= sen() cos() + cos() sen().
Logo:sen( ) =sen() cos() cos() sen() e
sen(2) =2 sen() cos().
Aula 18 Pr-Clculo 98
http://find/
