Download #nverse Trig 0unctions sin x, cos x, tan x, sec x

!nverse!Trig!*unctions
The!big!1!trigonometric!functions:
sin!x,!!!!cos!x,!!!!tan!x,!!!!sec!x
8ou!should!know!their!graphs,!special!values,!major!
identities,!and!their!derivatives.
@!table!!!learned!long!ago:
angle
sin!angle" cos!angle"
0
0
!#!0
2
1
"
!#!30F
D
1
1
!#! 2
2
3
2
"
!#!1GF
1
2
1
!#!
2
2
1
2
"
!#!D0F
3
3
2
1
2
"
!#!H0F
2
1
!#!1
2
0
page!2
Three!inverse!trig!functions!we!need!to!know!about:
!!!!!sin!1!x!#!arcsin!x,!!!cos!1!x!#!arccos!x!!!a!little",
tan!1!x!#!arctan!x
sin!1!x:!!the!angle!whose!sine!is!!x
"
!1! 1
ie!!sin
!#! !!!@IW@8K!radians"
2 D
!"#$%&'#(%)!%)*'+,%-.!-$'#(-(-'/0
sin!1!sin!x"!!#!!x!!!!!!!!!!!!!!!!!!sin!sin!1!x"!!#!!x
"
"
!if!!!! 2 !#!x!#! 2 !!!!!!!!!!!!!!!!!!!!!!if!!!!1!#!x!#!1
cos!1!cos!x"!!#!!x!!!!!!!!!!!!!!!!cos!cos!1!x"!!#!!x
!!!if!!0!#!x!#!"!!!!!!!!!!!!!!!!!!!!!!!!!!!if!!!1!#!x!#!1
tan!1!tan!x"!!#!!x!!!!!!!!!!!!!!!!tan!tan!1!x"!!#!!x
"
"
!!if!!! 2 !#!x!#! 2 !!!!!!!!!!!!!!!!!!!!!!!!for!every!!x
page!3
Triangles
cosM1!x
1
x
sinM1!x
"
!1
!1
!n!particular,!!sin !x!$!cos !x!#!
2
x
tanM1!x
1
page!1
1',-2%(-2'/!%#$!-#('*,%)/:
y!#!sin!1!x!!"!!x!#!sin!y,
d
dy
dy
1
!x!#!sin!y"!!"!!1!#!!cos!y"! !!"!! !#!
dx
dx
dx cos!y
Kince!!cos!y!#! 1!!!sin2!y !#! 1!!!x2 ,
d
!sin!1!x!!#!!
dx
#
backwards:!!!!!!! !
1
1!!!x2
1
1!!!x2
!dx!!#!!sin!1!x!!$!!c
y!#!cos!1!x!!"!!x!#!cos!y
d
dy
!x!#!cos!y"!!"!1!#!!!!sin!y"! !!"
dx
dx
dy d
1
!1!x!!#!!!! 1
!!!!! !#! !cos
!!#!!!!
dx dx
cos!y
1!!!x2
page!G
y!#!tan!1!x!!"!!x!#!tan!y,
d
dy
1
2!y"! dy
!x!#!tan!y"!!"!!1!#!!sec
!!"!! !#! 2
dx
dx
dx sec !y
Kince!!sec2!y!#!1!$!tan2!y!#!1!$!x2,!we!have
d
!1!x!!#!! 1
!tan
dx
1!$!x2
#
Nackwards:!! !
1
!1!x!$!c
!dx!!#!!tan
1!$!x2
Every!once!in!a!while:
d
!sec!1!x!!#!!
dx
x
#x
!!!!!!!!!!!!!!!!! !
1
x2!!!1
1
x2!!!1
,
!dx!#!sec!1!x!!$!!c