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