STUDY PACKAGE. Subject : Mathematics Topic : INVERSE TRIGONOMETRY. Available Online :

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2 . Principal Values & Domains of Inverse Trigonometric/Circular Functions: Function Domain Range (i) = sin where (ii) = cos where 0 (iii) = tan where R (iv) = cosec where or, 0 (v) = sec where or 0 ; (vi) = cot where R 0 < < NOTE: (a) st quadrant is common to the range of all the inverse functions. (b) rd quadrant is not used in inverse functions. (c) th quadrant is used in the clockwise direction i.e. 0. (d) No inverse function is periodic. (See the graphs on page 7) Solved Eample # Solution Find the value of tan Let = tan cos = tan 6 = tan 6 = cos tan Ans. tan Find the value of the followings :.. () sin sin Ans. () cosec [sec ( ) + cot ( )] Ans. Solved Eample # Find domain of sin ( ) Let = sin ( ) For to be defined ( ) 0 0 [, ] Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0 Find the domain of followings : () = sec ( + + )

3 Get Solution of These Packages & Learn b Video Tutorials on () = cos () = tan ( ) Answers () (, ] [, ] [0, ) () R () (, ] [, ). Properties of Inverse Trigonometric Functions: Propert - (A) (i) sin (sin ) =, (ii) cos (cos ) =, (iii) tan (tan ) =, R (iv) cot (cot ) =, R (v) sec (sec ) =,, (vi) cosec (cosec ) =,, These functions are equal to identit function in their whole domain which ma or ma not be R.(See the graphs on page 8) Solved Eample # Find the value of cosec cot cot Let = cosec cot cot cot (cot ) =, R cot cot = from equation (i), we get = cosec = Ans.. Find the value of each of the following : (6) cos sin sin Answers (6) Propert - (B) 6 (7) not defined...(i) (7) sin cos cos (i) sin (sin ) =, (ii) cos (cos ) = ; 0 (iii) tan (tan ) = ; (iv) cot (cot ) = ; 0 < < (v) sec (sec ) = ; 0, These are equal to identit function for a short interval of onl. (See the graphs on page 9-0) Solved Eample # Find the value of tan tan Let = tan tan Note tan (tan ) =, (vi) cosec (cosec ) = ; 0, Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

4 , tan tan, graph of = tan (tan ) is as : from the graph we can see that < <, then = tan (tan ) can be written as = = tan tan = solved Eample # Find the value of sin (sin7) Let = sin (sin 7) Note : sin (sin 7) 7 as 7, < 7 < graph of = sin (sin ) is as : From the graph we can see that then = sin (sin ) can be written as : = sin (sin 7) = 7 = Similarl we have to find sin (sin( )) then < < Successful People Replace the words like; "wish", "tr" & "should" with "I Will". Ineffective People don't. Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

5 from the graph of sin (sin ), we can sa that sin (sin( )) = + ( ) = (8) Find the value of cos (cos ) (9) Find sin (sin ), cos (cos), tan (tan ), cot (cot) for, Ans. (8) (9) sin - (sin) = ; cos (cos ) = ; tan (tan ) = ; cot (cot ) = Propert - (C) (i) sin () = sin, (ii) tan () = tan, R (iii) cos () = cos, (iv) cot () = cot, R The functions sin, tan and cosec are odd functions and rest are neither even nor odd. Solved Eample # 6 Let Find the value of cos {sin( )} = cos {sin( )} = cos ( sin ) cos ( ) = cos, = cos (sin ) = cos cos < < graph of cos (cos ) is as : from the graph we can see that then = cos (cos) can be written as = +...(i) from the graph cos cos = + = from equation (i), we get = = Ans. Find the value of the following 7 (0) cos ( cos ) () tan tan 8 () tan cot Successful People Replace the words like; "wish", "tr" & "should" with "I Will". Ineffective People don't. Answers. (0) () () 8 Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

6 Propert - (D) (i) cosec = sin ;, (ii) sec = cos ;, (iii) cot Solved Eample # 7 Solution tan tan Find the value of tan Let ; 0 ; 0 cot = tan cot cot ( ) = cot, R equation (i) can be written as = tan cot = tan cot cot = tan > 0 = tan tan = Find the value of the followings...(i) () sec cos () cosec sin Answers. () Propert - (E) () (i) sin + cos =, (ii) tan + cot =, R (iii) cosec + sec =, Solved Eample # 8 Find the value of sin (cos + sin ) when = Let = sin [cos + sin ] sin + cos =, = sin cos cos = sin cos = cos (cos ) = Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 6 of 0 = cos cos...(i) cos (cos ) = [, ]

7 [, ] cos cos = =. Solve the following equations () tan + cot = (6) sin = cos Answers. () = (6) = Propert - (F) (i) sin (cos ) = cos (sin ) = from equation (i), we get, (ii) tan (cot ) = cot (tan ) =, R, 0 (iii) cosec (sec ) = sec (cosec ) = Solved Eample # 9 Find the value of sin tan. Let = sin tan Note : To find we use sin(sin ) =, For this we convert tan in sin Let = tan sin (sin ) = sin, >...(i) tan = and 0, sin = 0, sin (sin ) = equation (ii) can be written as : = sin...(ii) = tan from equation (i), we get = sin sin = Solved Eample # 0 Find the value of tan cos Let = tan cos...(i) tan = sin Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 7 of 0

8 Let cos = 0, and cos = equation (i) becomes = tan cos tan = = cos tan = ± 0, 0, tan > 0 from equation (iii), we get tan = from equation (ii), we get = Solved Eample # = Ans. = sin = sin (sin ) = sin...(ii) Successful People Replace the words like; "wish", "tr" & "should" with "I Will". Ineffective People don't. 0, sin (sin ) =...(ii) ( )...(iii) Find the value of cos (cos + sin ) when = Let = cos [cos + sin ] Aliter : Let sin + cos =, = cos cos cos = cos cos = sin (cos ) = = sin cos sin (cos ) =, sin cos = = from equation (i), we get =...(i) cos = cos = and 0, Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 8 of 0

9 equation (ii) can be written as = sin = cos cos = sin Now equation (i) can be written as = sin sin...(iii) [, ] sin sin = from equation (iii), we get = Find the value of the followings : (7) tan cosec (9) sin cot Answers : (7) (8) 6 6 (8) sec cot 6 6 (0) tan tan (9). Identities of Addition and Substraction: A. (0) (i) sin + sin = sin, 0, 0 & ( + ) 7 7 = sin, 0, 0 & + > Note that: + 0 sin + sin + sin + sin (ii) cos + cos = cos, 0, 0 (iii) tan + tan = tan, > 0, > 0 & < Note that : B. = + tan, > 0, > 0 & > =, > 0, > 0 & = < 0 < tan + tan < ; > < tan + tan < (i) sin sin = sin, 0, 0 Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 9 of 0 (ii) cos cos = cos, 0, 0,

10 (iii) tan tan = tan, 0, 0 Note: For < 0 and < 0 these identities can be used with the help of properties (C) i.e. change and to and which are positive. Solved Eample # 8 Show that sin + sin = sin 7 8 > 0, > 0 and = > 7 sin + sin = sin Solved Eample # Evaluate: 8 = sin = sin 8 6 cos + sin tan 6 6 Let z = cos + sin tan 6 sin = cos z = cos + cos z = cos cos > 0, > 0 and < 6 tan. 6 6 tan 6 cos cos = cos equation (i) can be written as 6 6 z = cos tan z = sin tan sin = tan 6 6 from equation (ii), we get 6 6 z = tan tan (i) 6 69 = cos 6...(ii) z = 0 Ans. Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 0 of 0 Solved Eample #

11 Evaluate tan 9 + tan 9 > 0, > 0 and 9 > 9 tan 9 + tan = + tan 9. = + tan ( ) =. tan 9 + tan =. 6 () Evaluate sin + sin + sin 6 () If tan + tan = cot then find () Prove that cos Solve the following equations () tan () + tan () = Answers. () C. (i) sin = (ii) cos ( ) = (iii) tan (iv) sin (v) cos (See the graphs on page 0) Solved Eample # cot + cos = 6 9 () = 9 = = sin sin sin cos cos tan tan tan tan tan tan tan = tan 0 0 () sin + sin = () = 6 () = 0 0 Define = cos ( ) in terms of cos and also draw its graph. Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0 Let = cos ( ) Note Domain : [, ] and range : [0, ]

12 Let cos = [0, ] and = cos = cos ( cos cos ) = cos (cos )...(i) Fig.: Graph of cos (cos ) [0, ] [0, ] to define = cos (cos ), we consider the graph of cos (cos ) in the interval [0, ]. Now, from the above graph we can see that (i) 0 cos (cos ) = from equation (i), we get = = 0 = cos (ii) cos (cos ) = from equation (i), we get = = < = cos < (iii) cos (cos ) = + from equation (i), we get = + = + = + cos from (i), (ii) & (iii), we get cos = cos ( ) = cos cos Graph : For = cos ( ) domain : [, ] range : [0, ] (i), = cos. d = d ; ; ; = ( ) /...(i) Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0 d < 0 d, decreasing Successful People Replace, the words like; "wish", "tr" & "should" with "I Will". Ineffective People don't. again we dferentiate equation (i) w.r.t., we get

13 d d = ( ) / d < 0, d (ii) <, = cos. (iii) d = d increasing, d (a), 0 then < 0 d concavit downwards, d > 0 d, d and d concavit downwards, 0 d (b) 0, then > 0 d concavit upwards 0, d d Similarl < then < 0 and d > 0. d the graph of = cos ( ) is as = ) / ( (6) Define = sin ( ) in terms of sin and also draw its graph. (7) Define = tan Answers sin (6) = sin ( ) = sin sin graph of = sin ( ) in terms of tan and also draw its graph. ; ; ; Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

14 (7) = tan = D. tan tan tan ; ; ; Fig.: Graph of = tan z z If tan + tan + tan z = tan z z, > 0, > 0, z > 0 & ( + z + z) < NOTE: (i) If tan + tan + tan z = then + + z = z (ii) If tan + tan + tan z = then + z + z = (iii) tan + tan + tan = (iv) tan + tan + tan = Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

15 . (i) (iii) = sin,, O Inverse Trigonometric Functions Some Useful Graphs, = tan, R,,, O O (ii) (iv) = cos,, [0, ] O (v) = sec,, 0, U, (vi) = cosec,, = cot, R, (0, ) O O, 0 U 0, Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page of 0

16 Part - (A) (i) = sin (sin ) = cos (cos ) =, [, ], [, ]; is aperiodic )º O + (ii) = tan (tan - ) = cot (cot - ) =, R, R; is aperiodic )º O (iii) = cosec (cosec ) = sec (sec ) =,, ; is aperiodic = = = O = Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 6 of 0

17 Part -(B) (i) = sin (sin ), R,,, is periodic with period (ii) (iii) (iv) = ( + ) = cos (cos ), R, [0, ], is periodic with period )º O = = + = = = = O = tan (tan ), R ( n ), n, is periodic with period = + = = sec (sec ), is periodic with period ; R ( n ), n, 0, U, = + = + = O = O = = = = = Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 7 of 0

18 (v) = c o t (cot ), is periodic with period ; R {n, n }, 0,, (vi) = cosec (cosec ), is periodic with period ; R {n, n },, {0} Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 8 of 0

19 Part - (C) (i) graph of = sin (ii) graph of = cos ( ) Note : In this graph it is advisable not to check its derivabilit just b the inspection of the graph because it is dficult to judge from the graph that at = 0 there is a shapr corner or not. (iii) graph of = tan (iv) graph of = sin (v) graph of = cos Teko Classes, Maths : Suhag R. Karia (S. R. K. Sir), Bhopal Phone : , page 9 of 0