Identidades Trigonometric As Derivadas e Integrais
3
Identidades Trigonométricas, Derivadas e Integrais Identidades Trigonométricas e Outras Relações 01. 1 a cos a sen 2 2 = + 20. 2 b a cos 2 b a sen 2 b sen a sen − ⋅ + ⋅ = + 02. a cotg 1 a cos a sen a tg = = 21. 2 b a cos 2 b a sen 2 b sen a sen + ⋅ − ⋅ = − 03. a tg 1 a sen a cos a cotg = = 22. 2 b a cos 2 b a cos 2 b cos a cos − ⋅ + ⋅ = + 04. a cos 1 a sec = 23. 2 b a sen 2 b a sen 2 b cos a cos − ⋅ + ⋅ − = − 05. a sen 1 a cosec = 24. b cos a cos ) b a ( sen b tg a tg ⋅ ± = ± 06. 1 a sec a tg 2 2 − = 25. ) b a ( cos ) b a ( sen 2 b 2 sen a 2 sen − ⋅ + ⋅ = + 07. 1 a cosec a cotg 2 2 − = 26. ) b a ( cos ) b a ( sen 2 b 2 sen a 2 sen + ⋅ − ⋅ = − 08. a cos b sen b cos a sen ) b a ( sen ⋅ ± ⋅ = ± 27. ) b a ( cos ) b a ( cos 2 b 2 cos a 2 cos − ⋅ + ⋅ = + 09. b sen a sen b cos a cos ) b a ( cos ⋅ ⋅ = ± m 28. ) b a ( sen ) b a ( sen 2 b 2 cos a 2 cos − ⋅ + ⋅ − = − 10. b tg a tg 1 b tg a tg ) b a ( tg ⋅ ± = ± m 29. b 2 cos a 2 cos ) b a ( 2 sen b 2 tg a 2 tg ⋅ ± = ± 11. a cos a sen 2 a 2 sen ⋅ ⋅ = 30. )] b a ( cos ) b a ( [cos 2 1 b sen a sen − − + ⋅ − = ⋅ 12. a sen a cos a 2 cos 2 2 − = 31. )] b a ( cos ) b a ( [cos 2 1 b cos a cos − + + ⋅ = ⋅ 13. a tg 1 a tg 2 a 2 tg 2 − ⋅ = 32. )] b a ( sen ) b a ( sen [ 2 1 b cos a sen − + + ⋅ = ⋅ 14. ) a 2 cos 1 ( 2 1 a sen 2 − = 33. ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⋅ − − + ⋅ − = ⋅ b cos a cos ) b a ( cos ) b a ( cos 2 1 b tg a tg 15. ) a 2 cos 1 ( 2 1 a cos 2 + = 34. ) a ( sen arc a sen = α ⇔ = α 16. a 2 cos 1 a 2 cos 1 a tg 2 + − = 35. ) b ( cos arc b cos = α ⇔ = α 17. ) a cos 1 ( 2 1 2 a sen 2 − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 36. ) c ( tg arc c tg = α ⇔ = α 18. ) a cos 1 ( 2 1 2 a cos 2 + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 37. Ímpar Função a sen a) ( sen ⇒ − = − 19. a cos 1 a cos 1 2 a tg 2 + − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ 38. Par Função a cos a) ( os c ⇒ + = −
-
Upload
rodrigo-torres -
Category
Documents
-
view
32 -
download
1
Transcript of Identidades Trigonometric As Derivadas e Integrais
IIddeennttiiddaaddeess TTrriiggoonnoommééttrriiccaass,, DDeerriivvaaddaass ee IInntteeggrraaiiss
IIddeennttiiddaaddeess TTrriiggoonnoommééttrriiccaass ee OOuuttrraass RReellaaççõõeess
01. 1acosasen 22 =+ 20. 2
bacos2
basen2bsenasen −⋅
+⋅=+
02. acotg
1acosasenatg == 21.
2bacos
2basen2bsenasen +⋅
−⋅=−
03. atg
1asenacosacotg == 22.
2bacos
2bacos2bcosacos −⋅
+⋅=+
04. acos
1asec = 23. 2
basen2
basen2bcosacos −⋅
+⋅−=−
05. asen
1acosec = 24. bcosacos)ba(senbtgatg
⋅±
=±
06. 1asecatg 22 −= 25. )ba(cos)ba(sen2b2sena2sen −⋅+⋅=+
07. 1acosecacotg 22 −= 26. )ba(cos)ba(sen2b2sena2sen +⋅−⋅=−
08. acosbsenbcosasen)ba(sen ⋅±⋅=± 27. )ba(cos)ba(cos2b2cosa2cos −⋅+⋅=+
09. bsenasenbcosacos)ba(cos ⋅⋅=± m 28. )ba(sen)ba(sen2b2cosa2cos −⋅+⋅−=−
10. btgatg1
btgatg)ba(tg⋅
±=±
m 29.
b2cosa2cos)ba(2senb2tga2tg
⋅±
=±
11. acosasen2a2sen ⋅⋅= 30. )]ba(cos)ba([cos21bsenasen −−+⋅−=⋅
12. asenacosa2cos 22 −= 31. )]ba(cos)ba([cos21bcosacos −++⋅=⋅
13. atg1
atg2a2tg 2−⋅
= 32. )]ba(sen)ba(sen[21bcosasen −++⋅=⋅
14. )a2cos1(21asen2 −= 33. ⎥⎦
⎤⎢⎣⎡
⋅−−+
⋅−=⋅bcosacos
)ba(cos)ba(cos21btgatg
15. )a2cos1(21acos2 += 34. )a(senarcasen =α⇔=α
16. a2cos1a2cos1atg2
+−
= 35. )b(cosarcbcos =α⇔=α
17. )acos1(21
2asen2 −=⎟⎠⎞
⎜⎝⎛ 36. )c(tgarcctg =α⇔=α
18. )acos1(21
2acos2 +=⎟⎠⎞
⎜⎝⎛ 37. ÍmparFunçãoasena)(sen ⇒−=−
19. acos1acos1
2atg2
+−
=⎟⎠⎞
⎜⎝⎛ 38. ParFunçãoacosa)(osc ⇒+=−
RReeggrraass ddee DDeerriivvaaççããoo:: )u( ee )v( ssããoo FFuunnççõõeess ddee )x( DDeerriivváávveeiiss,, )n( ee )a( ssããoo CCoonnssttaanntteess
01. ay = 0'y = 19. usenarcy = 2u1
'u'y−
=
02. nuy = )1n(,'uun'y 1n −≠⋅⋅= − 20. ucosarcy = 2u1
'u'y−
−=
03. nuay ⋅= )1n('uuna'y 1n −≠⋅⋅⋅= − 21. utgarcy = 2u1'u'y
+=
04. vuy += 'v'u'y += 22. utgcoarcy = 2u1'u'y
+−=
05. vuy −= 'v'u'y −= 23. )1|u|(,uecsarcy ≥= )1|u|(,1u|u|
'u'y2
>−⋅
=
06. vuy ⋅= 'vuv'u'y ⋅+⋅= 24. 1)|u|(,ucosecarcy ≥= )1|u|(,1u|u|
'u'y2
>−⋅
−=
07. vuy = 2v
'vuv'u'y ⋅−⋅= 25. usenhy = ucosh'u'y ⋅=
08. ua'y = )0ae1a(,aln'ua'y u >≠⋅⋅= 26. ucoshy = usenh'u'y ⋅=
09. uey = 'ue'y u ⋅= 27. utghy = uhsec'u'y 2⋅=
10. ulog'y a= )0ae1a(,elogu'u'y a >≠⋅= 28. ucotghy = ucosech'u'y 2⋅−=
11. ulny = u'u'y = 29. uechsy = utghuhsec'u'y ⋅⋅−=
12. vuy = uln'vu 'uuv'y v1v ⋅⋅+⋅⋅= − 30. ucosechy = ucotghucosechu'y' ⋅⋅−=
13. useny = ucos'u'y ⋅= 31. usenhargy = 1u
'u'y2 +
=
14. ucosy = usen'u'y ⋅−= 32. ucoshargy = )1u(,1u
'u'y2
>−
=
15. utgy = usec'u'y 2⋅= 33. utghargy = )1|u|(,u1'u'y 2 <
−=
16. ucotgy = ucosec'u'y 2⋅−= 34. utghcoargy = )1|u|(,u1'u'y 2 >
−=
17. uecsy = utgusec'u'y ⋅⋅= 35. uechsargy = )1u0(,u1u
'u'y2
<<−⋅
−=
18. ucosecy = ucotgucosecu'y' ⋅⋅−= 36. ucosechargy = )0u(,u1|u|
'u'y2
≠+⋅
−=
IInntteeggrraaiiss IImmeeddiiaattaass ee FFóórrmmuullaass ddee RReeccoorrrrêênncciiaa:: )u( ee )v( ssããoo FFuunnççõõeess ddee )x( ,, )n( ,, )a( ee )C( ssããoo CCoonnssttaanntteess
01. Cudu +=∫ 19. Cauulnau
du 22
22+−+=
−∫
02. )1n(,C1n
uduu1n
n −≠++
=+
∫ 20. Causecarc
a1
auudu
22+=
−∫
03. C|u|lnu
du+=∫ 21. )au(,C
ausenarc
uadu 22
22<+=
−∫
04. )0ae1a(,Caln
aduau
u >≠+=∫ 22. Cu
uaalna1
uaudu 22
22+
++−=
+∫
05. Cedue uu +=∫ 23. Cu
uaalna1
uaudu 22
22+
−+−=
−∫
06. Cucosduusen +−=∫ 24. Cucoshduusenh +=∫
07. Cusenduucos +=∫ 25. Cusenhduucosh +=∫
08. C|usec|lnduutg +=∫ 26. Cutghduusech2 +=∫
09. C|usen|lnduucotg +=∫ 27. Cutghcoduucosech2 +−=∫
10. C|utgusec|lnduusec ++=∫ 28. Cusechduutghusech +−=⋅∫
11. C|ucotgucosec|lnduucosec +−=∫ 29. Cucosechduucotghucosech +−=⋅∫
12. Cutgduusec2 +=∫ 30. duuasenn
1nan
uacosuasenduuasen 2n1n
n ∫∫ −− −
+⋅
−=
13. Cucotgduucosec2 +−=∫ 31. duuaoscn
1nan
uaoscuaensduuaosc 2n1n
n ∫∫ −− −
+⋅
=
14. Cusecduutgusec +=⋅∫ 32. duuagt)1n(auagtduuagt 2n
1nn ∫∫ −
−
−−
=
15. Cucosecduucotgucosec +−=⋅∫ 33. duuagcot)1n(auagtcoduuagcot 2n
1nn ∫∫ −
−
−−
−=
16. Cautgarc
a1
audu
22 +=+∫ 34. duuasec
1n2n
1)(nauatguasecduuasec 2n
2nn ∫∫ −
−
−−
+−⋅
=
17. )au(,Cauauln
a21
audu 22
22 >++−
=−∫ 35. duuacosec
1n2n
1)(nauacotguacosecduuacosec 2n
2nn ∫∫ −
−
−−
+−⋅
−=
18. Cauulnau
du 22
22+++=
+∫ 36. ∫∫ −
−
+−⋅−++⋅
=+ 1n222
n122
n22 )au(du
)1n(a23n2)au(u
)au(du
