Inverse-Trigonometric-Functions.pdf

Trigonometri formula

1. Complete Study Guide & Notes On
INVERSE TRIGONOMETRIC FUNCTIONS

uide By OP Gupta (Indira Award Winner)

The sen of Mathematics not o mal simple
Gxt make compat hinge mpl!

IMPORTANT TERMS, DEFINITIONS & RESULTS

‘ A list of rmac for Trigonometric Funetions (lass XI) has been added for
reference at the end ofthis om guide.

complicated

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and on) The mes of he Anton fis denoted by $: BA such at (92 (Oe 3
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02 List of Formulae and thr proof for rasé gérer Funan
a asien!) y o fl

dcos (x)=se0" (4) xf,

«(Mags

sta A Le Ce
e O rar) co
PROOF 4) Lot sil (> 0 then, SnB= ©
RP:
ie mr
ter reg Roti ie Same way!
Be asin Cox) sin“, real] bcosr(-x)= #00 1x xe[-11]
A Teen owe (= cou h, et
sec (x)= n-secs, |xi2l Hoot"!(-x)=n-cot" x, xER
PROOF D) La cor 4x) 0 then 060
2 “ans orge o
Soon Ser) mr
Sin cnprove terest
©. asin(sinz) hera bycos(cosx)=x, 05x"

ou (ns)=5, Foro

see" (seox)=x, 05 v5.8, Hoot Meats) =x, Dev

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2. Inverse Trigonometric Functions By OP Gupta (INDIRA AWARD Winner, Elect. & Comm, Engineering)

PROOF a) Lat a (+) 0)
then, sin@= (a)
Substituting the value of in () from i) we get,
sin" (Replacing 0 bys)
sin sin = mue

Fe er els we can proceed nd
D asar weft]

bptan x + cot

ocosec-tr+se tx= À ixjalto, xt orxet
PROOF a) Let sin”#()= 0 then, sind= x

Be

10) P Poco yo.

ran

)

and sin! y =P. Then, sind =x and sinp= y.
bcos + coe sin

ou an (EE

aa,

PROOF a) Let an tx
Now sin +
=sin@+p)

EX

rer]
FAN mes

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sin! y=

3. „A Complete Formulae Guide $ Compiled By OP Gupta (M.+91-9650350480 | [phone])_
Trace [order]
Doyo Proc in)
9 Let tan"! =6 and tan”! y =P. Then, tan = x and tanB=
Now tan(0+p)-

i tandtmp~ 1-9

cope [EE

= tan ta

For x>0, tax will be a positivo angle and for y>0, tan» will also be a postive a
‘Therefore, LHS of (c) will be a positive angle and hence RHS should also be a positive angle.

Case! When 40,9 0 and at then E spose
AN
Hence, tan x + tn! y= ta &

so eel) ame anos
Hence, tan x. i (Py
Case Wn T0 ana > © ose.
s we), 4 ms 22 tt pi a hee
N
: ma

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4. Cas II When x>0,y<0 and ay<-1 then

zoe

is negativo. Also if x>0,y<0 then,

in
(0.2) 5 0-Pe (0x) ie, ta

tte (x)

As [Ez] is a negative angle. Therefore we add in RHS of (A) lo make it postive and
balanced.
ene, tata! ran] oa y<o and mr]

Wen x<0,y>0 and pci then EZ is postive, Abo if x<0,y>0 then,

8) 50(08) 0. 0e(-$0)-pe(-Z9)
¿>

>0 and mr]

oe

: org Aad
So, ent willbe a negative angle and tan SA ir

ae two vil head RSA)

Hence, tan x tan

E
mr]

other proved inthe same way!

05. Princip Value: Numerically smallest angle is know as the principal value

Finding the pricipal value: For finding the principal value, following algoritun can be followed

STEPI- Firstly, draw tigonometrc circle and mark the quadrant in which the angle may ie.

STEP2- Select anticlockwise direction for 1% and 2° quadrants and clockwise direction for 3° and
5 quadrants

STEPS- Find the angles in the first rotation

STEP Select the mumeriall least (magnitude wis) angle among these two values. The angle Uns
found willbe the principal value.

STEPS- In case, two angles one with positive sign and the other with the negative sign qualify for
the numerically least angle then ts the convention to select the angle with positive sign as
principal value

The principal value ts never mnerically greater than #

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5. A Complete Formulae Guide % Compiled By OP Gupta (M+91-9650350480 | [phone])

DE Table demonstrating domalns and ranges of Inverse Trigonometric functions

cots R

Discussion about the range of inverse circular funcions,
‘other than thelr respective ieineipal value branch!

We know that the domain of sine Fiction these cal uber anaes the closed
interval [-1, 1] If we reset its sgn >: [3 3 5 rc. then, it
PO) à

bijective with ie range. TA) Sop te can dale Inverso of sine Function in
each ofthese intervals. Hee Ale MR als ob Y functión except principal value

Branch (here except of fr sor Ann BA as the range of sin’ other

7%
than its princifál Yátue branch The(Sne discusión can be extended for ether inverse
circular functions (Refer Q16 in Malhematicia Nol.| By OP Gupta)

05. To siaplifsterse rigonametrical expressions following substations can be considered

‘alan of x= ac0t0

x= asind or x= acox0

x= asco ar x= a cose)

x= 400820

ya 00620

x= asin’@ or x=acos'0

x= ata O of x= acot'®

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6. Inverse Trigonometric Functions By OP Gupta (INDIRA AWARD Winner, Elect. & Com
‘ove te Jollowings and keep hem in mind
2 The symbol sx is used o denote the smallest angle whether positive or negative, such
‘that the sine of tis angle will give us x. Similarly cas x, nx, cosee x, sec, and
cot are defined.
2 Fou should note tht sin" con be writen as aresins, Similarly other Inverse Trigonomenrc
Funetions can also be writen as arecosx arc arcseex ec.

Engineering)

(and similrly other Inverse Trigonometric Functions) is entirely
diferent from (sin 9". In fact, sine 18 the measure of an angle in Radians whose sine is x

las (i) foc bi pere gon)
2 Ko in fa thm im gemein er o nds

‘hey are valid only for some values of for which inverse migonamerte functions are well
defined!

O sn

+ Trigonometric Formulae (Only Ft ce}:

+ Relation between trigonometric rat
a) tano= an D 0-1
Ex 5

coso 1
coro = S080 ©) cose ge } so
+ nen, Q sap le ms eng 24 44
q 9 isin 2A= Dein des
bj see?
Ant
9 änd=2unen
e) cosectó=1 y, ES E
suai A | cordon A
rue
A Sn dcosB 4 dun
bjeos( A+B) = cos Acos BFsinAsin B eco 24 = 2608 4-1
Oem(4 Be B)=e dans [200842180124
cos’ B- sin" A 96024 =1-2sin’ A
4 sina B)sn(~B)= in’ ds B GRETA
— panama
ovina aca ea
indi aaa
: Preven ar
ala) ‘cot B + cot A

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7. A Complete Formulae Guide % Compiled By OP Gupta (M.+91-9650350480 | [phone])
IE]

C+D „ED

banc sin D

bun24= 2004
Paw

cheosC + 08D = 2008

Dsin34= Bain A- Asin! A

Jana
tanz JA tn A 6) 2sin AcosB =sin( A+ B)+ sin(A~
ara a ld
9 Transformation ofsums /dierences nto produces | DIOS ASIMB = n(A+ B)-sn(A 8)
e ricesersa 8) 2005 AcosB = cos( A+ B)+ cos(A-B)

a) sinC +sinD = 2sin + 200$ DP

1) 2sin Asin B= cos( 4~ B)-cos(4+B)

+ Relations in Different Measures of Angle

2 Angle in Radian Measure = (Angle in D

a
Messe)
EN

9 Angle in Degree Measure = (Angle in Radic n Mie)

A 0 flows ae impagos Mai

2 1Right angle = 90° Sacha oo
or > 0.0174 Fáljans(Apprax.) D bien 1745° or 206265 seconds

+ GenerdlSotitons
@) sins y À x= nr hoj nee ne z

D cosx=cosy => Binz y shore ne
9 tanx=tany > elo By, where ne Z

he.
@

© In actual practice, we omit the exponent ‘€’ and instead of writing À we simply write and
sinilarly for others.

er

For the complete discussion ofthe Trigonometic Functions, please refe o the
FORMULAE GUIDE Of TRIGONOMETRIC FUNCTIONS of Class XI

+ For the values of Trigonomerie Ratios at Standard Angles (Le, 0.30°,15,60° and 90°), check
following page.

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8. Inverse Trigonometric Functions By OP Gupta (INDIRA AWARD Winn

+ Trigonometric Ratio of Standard Angles

e |» Is To | om
o 8 E
5 B
sin : ES 1
con 1 L °
5 ° 1 5
in 4 4
cosec 2 E E 1
“| =
sec 1 El E 2 on
MEN
cot Pr 5 JE °
Re AA a
+ Frigonontric ios of Aid Angles
zo | zo QU E eerie
E ¿yy :
= ET a= Sat = asd | ano | an
En Ze Bo [ane | a0 emo
Lota om
‘al SB
GS AR [= ces fn | seco | eso
cone tl E onsco |= seco [= sexo | coso | co

+ Domain and Rangéo) Trigonametric Functions

| ® run
ES | R en
tan [tren nen R
cot | éeRxenmnez) R
wer | weriseunnen R-CLD
secx [er xe(n+l)a/2.neZ}| R=)

‘Any queries and/or suggestions), please write to me at
[email protected]
Please mention your details: Name, Sudent/Teacher/Tutor,
Shool/hstition Place, Contact No, you wish)

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