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Chapter 13 — Inverse Trigonometric Functions Inverse Function If y= f(x) and x=g(y) are two functions such that f(g (y))=y and g(f(y))=%, then f and g are said to be inverse of each other, ie. g= fi. lf y= f(x) then x= f(y). Inverse Trigonometric Functions As we know that trigonometric functions are not one-one and onto in their natural domain and range, so their inverse do not exist but if we restrict their domain and range, then their inverse may exists. Domain and Range of Inverse Trigonometric Functions The range of trigonometric functions becomes the domain of inverse trigonometric functions and restricted domain of trigonometric functions becomes range or principal value branch of inverse trigonometric functions. : Table for Domain, Range and Other Possible Range of Inverse Trigonometric Functions Principal value Fi . . Ly ‘unction Domain branch (Range) Other possible range y=sin?x — [-1,]] [-2 a =3n =] E x] te 2'2 2’ 2 2' 2 y=cos x [-1, J (0, n] [-1, OJ [7, 2m] etc. ee Se y= tan? x R (-£ x) (= +) (z 5) etc. 2'2 2’ 2/)\2' 2 Inverse Trigonometric Functions é 141 a ween Eee OOOO eee eed fu x .,_Principalvalue | Function Domain braneht (RARBE Other possible range. y=sec'x R-(-1,1) e nt] ~ eal in, 0] ans a [x, 2n] - -{% "ac (0, m) (-7, 0), (m, 2m) etc. y=cot x R " EXPERT Scale Up Graphs of Inverse Trigonometric Functions The graphs of inverse trigonometric functions with respect to line y=x are given in the following table P ; Graph : : z Graph : Function ~~ @y interchanging axes) _ = (By ‘mirror image) y=sin x ; % 7 Ft. 2) BR \ » Inverse Trigonometric Functions 143 PRIOR IRIE Nees S fl - RRARARRAR Aen f Graph Function + Ph ely (By interchanging axes) » (By mirror image) y= cosec™!x Y ifr no 6 branch of of an ninve igonometric C ‘the’ principal yalue’ branchiis te ken for the inverse trigonometric | “function, : : Elementary Properties of Inverse Trigonometric Functions Property | ™ (i) sin™ 1 (sin 0) = 0; oe] -] (ii) cos" (cos 0) = 0; 8 ¢[0, 7] (iii) tan“ (tan 0) = 0; 0 € (- ei s) Dibincrccninanenaasioe j } ee eee ee a at all al at il (iv) cosec™! (cosec 0) = 0;0 € - a I 0+#0 (v) sec" (sec 8) = 0; 0 €[0, x], 0# . (vi) cot™! (cot 6) = 6; 6 €(0, x) Property II (i) sin (sin“! x) =x;x €[-1,1] (ii) cos (cos x)=x;xe[-1,]] (iii) tan (tan x)=x;xeR (iv) cosec (cosec” 'x) =x;x E(—00 7-1) U(1, 0) (v) sec (sec? De x;x €(—0,-1) U(1, 0) (vi) cot (cot x)=x;xeR Property m (i) sin (— x) =- sin™ ‘asx e[- 1,1] (ii) cos” iC x)=n-cos x; x €[=1,1] (iii) tan“1(— x) =— tan™ TexeR (iv) cosec”! (-x) =— cosec™!x; x e(—», -1] U[1, ) (v). sec! (—x) =2—-sec! x; x €(— 0, -1] U[], ») (vi) cot! (-x)=n- cot x;xeR _ Property IV (i) sin™ (2) cosec tx; x €(- 0, —1] U[1, ~) (ii) cos? (; )asectxr x e(-@,-11U[L 2) (ii) tn(2)-| cot x, ifx>0 x —n+cot'x, ifx<0 Property V (i) sin x + cos! x =F x e[-1,1] (ii) tan x + cotx=2xeER (iii) sec? x + cosec™!x = =F xe(—%,-1] U[1, 0) LON Nl Ne NA NS / jis / tf. Property VIII (i) tan x + tan? y tan™ {z x} if xy <1 1=xy y L) ifx >0,y>0 and zy >1 =4 m+tan™ (5 —n+tan! a ifx <0,y <0 and xy >1 1-xy (ii) tan? x - tan7 y wn Z4} ifxy>-1 = retan( Zt if x >0, y <0 and xy <-1 1+ xy —n+ wn EX) ifx <0,y>0 and xy <-1 1+ xy Gg Knowledge Booster*> =) > tan” x, + tan“ x)4..4tan-!x, = tant | Si783 +5 = t= ~S24S4- =S¢+ Sk denotes hess sum n of the ‘products: of Xi XDn- Xp n takes KG at a time. = Where: Property IX x =cos}./1-x? = tan =sec? 7 =cosec “(2], x €(0,1) j1—x? « x=sin!J1—x? =tan™ =cot! —*__ = sec) =cosec™! ( d ; } x €(0,1) x x -1 (i) sin -1 (ii) cos (iii) tan7! x = on{ ats) =cos™! ate] =cot? f ) 1+x l+x x -1 l+x = cosec q ‘Competition Insight \ sin (2x Jl-x2), if- = 1 n+ tan"( 222) ifx<-1 i Property X sin (3x — 4x3), it-tex 2n+ cos '(4x° -3x), if-1 — a "| 4-38x “7 3 —n+ tan! ax = | ifx<-— L . 1 -3x? V3 
