inverse trigonometricbau/w21.10a/inverse_trig.pdf · 2021. 2. 24. · inverse trigonometric...
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Inverse trigonometric functions and their derivatives
Idea ! trigonometric functions have nice geometric
interpretations and so it is important to
understand their derivatives.
What about the
inverses of the trigonometric functions and
their derivatives ?
First : do we even have an inverse ?
In...
No ! Fails horizontal line test.
End of lecture
Just kidding !
~ in...
is"" iii. Ties:::* .
line test
An inverse f-'to a function f means
f-'
Cx ) -- y ⇐ fly ) -_x
In the case of sin,
sin- '
Cx ) =y ⇐ sincykx AND yet-E. I ]
Domain of sin-'
Cx ) : E- 1,1] r
range of sink : e- ¥ .¥] ÷f÷÷"""
Important : sin-'
Cx) 't sing,-
- sink)' '
'
÷¥i÷""
Il - sin (x)
I /! sin- '
Cx)
l l
÷.
"""
, ⇒
:/ it
-
- it
-I Iz E )2
Some notation :
sin- '
Cx) = arcsincx)
Example : compute sin- '
C -'
z) and
cos Carcsincrz ))Ans :
Some notation :
sin- '
Cx) = arcsincx)
Example : compute sin- '
C -'
z) and
cos Carcsincrz ))Ans : sin
- '
C -E) =-
E
ceresin ( E) = ¥ ⇒ coscarcsin ( OE )) -- cos (F) = I
Example : what is tancarcsin ¥ ) ?
Ans :
Example : what is tancarcsin ¥ ) ?
Ans'
.
if x=arcsinC¥),
then
sin Cx) = ¥
I ⇒ ?'
-112=7'
⇒ ? -748
⇒ ? = 548
7
€11 ⇒ tank) -- ÷48T
Reminder : since sin and sin"
are inverses
i sink)defined in the region,
÷:c:÷÷:÷: ::÷÷¥÷
Derivative ? Use chain rule.
if fcx)-
- sink), gcx) = sin
- '
Cx )
then fcgcx)) - x and ⇐ [ fcglxl) ]=¥ExJ=l
ddx-L-fcgh.tl ] -
- f 'CgCxDg'Cx)
('
Cgcx)) = coscarcsincx ))
if f - arcsinlx),
then sinCf) -- X and×
? - E
and costal -
- Fi so TFF g'G) =/ ⇒ g'G) =¥ for xc.tl-l )
Example : if fcx) = sin- '
( x"- l )
,
(a) find the domain of (Cx )(b) find L'Cx )(c) find the domain of L'Cx )
Ans :
Example : if fcx) = sin- '
( x"- l )
,
(a) find the domain of (Cx )(b) find L'Cx )(c) find the domain of L'Cx )
Ans : need x"- I C- E-1,1 ] or - lex
"- let
meaning 0*42-2 ⇒ -
"
rzsxs TE
Tdomain of flex )
Example : if fcx) = sin- '
( x"- l )
,
(a) find the domain of (Cx )(b) find L'Cx )(c) find the domain of L'Cx )
Ans : fix)= sin"
Cx "- I ) = sin-
tgcxl)
( txt-
- gtx) = ,¥⇒.
(4×3)
Example : if fcx) = sin- '
( x"- I )
,
(a) find the domain of (Cx )(b) find L'Cx )(c) find the domain of L'Cx )
Ans : f'Cx) -- 1-Miz(4×3)
I - ( x"- l)'
> O or 44-15<1 ⇐ - Is x"- I < I
⇐ Osx"s 2
⇐ xe C- 452,0)U(0,452 )
Other functions : inverse cosine
cos- '
Cx) -_ y ⇐ Cosey ) -_ x AND OEYET
£×[ cos- '
Cx) ]= ¥2 ,
- lcxcl
a:*:÷⇒ :*:÷Notation ! cos
- '
Cx ) -_ arccoscx)
/ ,C- ' 'T)
find out !
(0,1) l.I d
⇒ co,?)
I•
Costacos
- '
(x )÷.
I • (IT,- l )
O T
Other functions : inverse tangenttan
- '
Cx) -- y ⇐ fancy ) -_ x AND-
Icy 5¥
i.
.
-
Iz'
E E'E
-E tank) ILNotation : tan
- '
Cx ) -_ arctancx)
- -- --- - - - - - - -
- II z
- fix¥: ⇒
i ' ll l
- -- - - - - -
-
-
I-I tank) Iz 2
2
"m tank) -- as mytan
- '
Cx) -- Ex -512
fine tank)-
- - o xhjm.-
tank) ---
I
Example : simplify the expressions sin ( tan- '
Cx) )
Ans :
Example : simplify the expression sin ( tan- '
Cx) )
Ans : ify
-
- tan- '
Cx ),
then Yancy ) -_ x
and arx ⇒ ? = Fit
so sin ( tank ) ) -- sin Cy ) -- Xix
Example : evaluate Ising. arctan (¥)
Ans :
Example : evaluate Ising. arctan (¥)
Ans : Iim l
x-75IF
= - cs
y(÷, ¥7. arctancx.tl
-fimaoarctancy,= - I2
5- 3
Example : differentiate fcxl -- +1¥, and
gCxI= Xarctancex)
Ans :
Example : differentiate fcxl -- +1¥, and
gCxI= Xarctancex)
Ans : fat . Ctancx))"
(Cx)-
-
- lctancxl)"
# Itai 'CxD
=I(tan' 'Cx))
' ¥
Example : differentiate fcxl -- +1¥, and
gCxI= Xarctancex)
Ans : gcx) -
- X - arctancex)
g'Cx)-
- 1. arctancex) xx - axdtiarctancex) ]
= arctancex) + xg¥Ee×]
= arctancex ) tIt @
2x
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