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INVERSE TRIGONOMETRIC, RATIOS —, , , , , Introduction, , An equation of circle x+y’ =r is satisfied by trigonometric, function by substituting x = r cos 0, y =r sin 6. Thus trigonometric, functions ‘are also known as circular functions, we have studied, tigonometric ratios of various types of angles such that A + B, 2A,, , A, 3A, > etc., , We can solve easily the equations of type, , sin 90°=x, cos 180°=y, tan 135° =z etc., , It is clear that x = 1, y=- 1,2 =~ 1 ie. solution of each of the, tbove equation is unique. Now consider, , sing = cos a= 0, tan B=-3, , 2 2, , What are the solutions of ®, % B ? Are they unique ? In this chapter, We are going to find out the answers of such questions. We have to, obtain angles whose trigonometric ratios are known ‘that is rather, 'eVerse procedure. That is the reason of why these functions are called, , 'NVerse circular functions., , 7-D

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Inverse, , Basic Mathematics, 72 | Definition, If sin 9 = x then @ is called sine inverse of x and jg inc,, g=sin x $s h, Thus, ifsin @ =x then 8 = sin x, Similarly Ifcos 8= then @=cos "x, ; If tan @ =x then @=tan x, If cot @=x then @=cot x, Ifsec 8=x then O@=sec™x, Ifcosec@=x then =cosec™'x, , ‘ + =1. =I -1 aj, The functions sin “x, cos x, tan x, cot x and sec * and cosee', are called inverse trigonometric functions., 1, , =, N =, iote that sin’ x # Gnx, , General and principal values of inverse circular functions,, Trigonometric ratios are periodic with period 2nK., , -.sin (2nK + 8) = sin @, cos (2nK + @)=cos 0, ° K=1,23.., 1. ” Ifsin @ = x, then sin (2nK + 8) =x, @ =sin™'x, 2mK +0=sin'x, K=123, , eee he 2), “. Sin" x = (8/ sin 6 = x} are called general values of sin’ 't, , , , see, If > <6 2 then G is called principal value of sin’? x., €g. sin30° = 5, 30° = sin™(2) Princ in, ~ Principal value of sin >, &g. sin 159° = i, 2, , Ty 11 sees, , nical Publications”, , An up thrust for knowledge, , , , , , , , Then, , 3 Tf, Then, , Then, , Then, , 6& If, , Then Cosec”' x, , 0 ~ af(l), 150" = sin @) - General value of sin ' ;, afl, sin 'G) = 30°, 150° etc., cos@ = x, =i, Cos x = {0 eos 8 = x} are called general values of, cos x. If 0<@<7z then @ is called, principal value of cos | x., tan@ = x, Tan'x = {0 / tan 6 = x} are called general values of, tan’! x and if S" <0<5 then 9 is called, principal value of tan’' x., cot@ = x, Cot''x = {@/cot 6 =x} are called general values of, cot ' x and if = <os5 then 6 is, called principal value of cot" Bg, secO = x, Sec''x = {0/sec @ =x} are called general values of, sec’ x and if 00 <7then @ is called, principal value of sec” ‘x., cosecO@ = x, = {0/cosec @ = x} are called general, 3 “p ok x, values of cosec ' x and if = <0 >, , then 0 is called principal value of, cosec ' x., , SE, , ™, Technical Publications - An up thrust for knowledge, , Se 0 apr

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7-4 Inverse, , , , fathematics, Basic M vincipal Values of Inverse Functions, , Consider A, sin 45° “5 eg ca a, sin 135° = a .sin! (a) = 135°, sin 405° = a + sin" @) ror, Thus we have, , iv"() = 45,135, 4058, sin Gs = 45°, 135°, 405%, ......,, at Ga) is not unique value function, but it is Mhultivalag, function., From application point of view the smallest of these Values js, , important. Once this value is known other values can be written dom, immediately. This value is called the principal value., , Thus the principal value of sin” GG) = 45° or ©, , 2 4, , Properties of Inverse Function OR Relation, between Inverse Tri jonometric Functions, , 7.4, , , , :, \ perty1: 1) sit x= cosae""(Z); 1x14, and cosec”'y = gin! (x) x|21, —, , + er rnrneineiiiier ec, , Technical Publications™, , ~An up thrust for knowledge, , , , , , 7-5, 1 -f1, 2) cos’ x= sec (z)ree1, sec 'x= cos” (1, xyilxiea, e a, Ayan x=00t' (2):x>0, and cot"x = tan” (x) x0, , A afi, and cot x-n=tan"' (1)., m= tan («<0, , , , , , , sin'x = 0, , x = sin@ --- [By definition of inverse function], a, ede : 1, , a eonen el --- |sin = 8, Lat, , cosec® = >, 6 = coses"(2) [By definition of inverse function], oy afl, , sin"x = cosec'(=, , Similarly, we can prove, , 2) Let, , , , T, , 4 (1, cosec "x = sin, , 8., cos x = 6, , I, , x cos 8, , aie, sec 8, , x, , ~. By invertendo, , Technical Publications » -An up thrust for knowledge, , +++ |cos 8, , sec 8

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y Ca “ x = -sin®, x = sin(-6), a8, -6 = sin’ x [By definition of inverse function}, @ = -sin'x, sin'(-x) = ~sin'x, similarly, we can Prove ‘ gyLet cos (-x) = 9, ot * coe"(?) -x = cos®, . n x = —cos®, 3)let tnx = 8 ., ase o x = cos(180°-6) .. |cos (180° ~ 8) = — cos 9), 1 1 x = cos(m-—8), =— +. {tan @=—] ~, * = cot @ cotd . m-@ = cos'x [By definition of inverse function], oe z @ = n-cos'x, x . -1 Mbs -1, : s.cos (—x) = M—cos x, e= oot" (2) 3)Let tan '(—x) = 0, vata ga ) & -x = tan@ [By definition of inverse function}, s x = -tan@, Similarly,, ly, We can prove ’ x = tan(-0) [tan (— 6) = tan 6], cot'x = tan’/— x= tn 5 -6 = tan'x [By definition of inverse function}, Property 2. sy4in“(-x)=~sin*x |x| <4 2 ; .% ras, 2) 008"(-x) = n- cosy 1x “tan” (-x) = tan x, i Is1 :, 3) tan'(-x)=-tany xe R Similarly, we can prove that :, Proof : ) cot '(-x) = n-cot' (x) xeR, YL sin" (-x) = 9 5) sec"'(—x) = m-sec™' (x) |x|21, 6) cosec™ -t, Bree * sec (~x) = —cosec (x) [x21, mae [By definition of inverse functio#

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Basic Mathematics, , , , Property 3: 1) sin''(sin®)=0;-5 < 0 so, , , , 2) cos‘ (cos @)=6;0<0<n, , , , , a ei 8 x, 3) tan” (tan @)=0;-5 < 0 < F, , , , , 4) cot (cot e)=0;0<0<n, , , , 5) sec'(sec8)=0;0< 0 < 79 #f, , , , 8) cosec(cosec 8)=8;-F < 9 < 3.0 x0, , , , Proof :, 1) Let sin® = x, @=sin'x .. [By definition of inverse Function], 8 = sin ' (sing), orga bes -»- be=sing, *. sin’ (sin@) = 9, Similarly,, , in this Way We can prove all the results of Property 3., , , , , , , Property 4: Yysin sin" x) =x; —4 Sx<1, 2) cos (cos x) =x: _ 4 Sxs<1, 3) tan (tan x)=x:xe R, , 4) cot (cot )=x:xeR, , , , , 5) sec(sec'x)=x-y 5 4, x2 9, , $) cosec (cosec'x) =x: » Wx<, , , , , , , , Let sinty 2 g, , sin8 [By definition of inverse functio!, , alii, , ~AN up thrust for knowledge, , , , asic Mathematics :, , , , x = sin (sin x), , “sin (sin! x)= x, , Similarly, in this way we can prove all, the results of, Property 4,, , , , , 4 &, , Property 8: t)-sin’ x + cos 'x =% Ix|<4, 2) tan"x+ cot 'x = % KER, 3) sec'"x + cosec"'x =¥ Ixle4, , , , , , , , , , , Proof :, 1) Let sin'x = 9, x=, x=, 2 t, cos'x = 7-8 [By definition of inverse function}, ®+cos'x = 5, sin’ x + cog? =}, , 2) Let tan'x = 9, , , , , , , , , , xX = tand [By dc‘inition of inverse function], x= cot (8 ) cot (3-0) = tnd, , “ cot!x = 5-0 [By definition of inverse function], , % fe ae, , “8+ cot'x = 3