INVERSE TRIGONOMETRIC FUNCTIONS

Transcription

1 Chapter INVERSE TRIGONOMETRIC FUNCTIONS Overview Inverse function Inverse of a function f eists, if the function is one-one and onto, ie, bijective Since trigonometric functions are many-one over their domains, we restrict their domains and co-domains in order to make them one-one and onto and then find their inverse The domains and ranges (principal value branches) of inverse trigonometric functions are given below: Functions Domain Range (Principal value branches) y sin [,], y cos [,] [0,] y cosec R (,), {0} y sec R (,) [0,] y R, y cot R (0,) Notes: (i) The symbol sin should not be confused with (sin) Infact sin is an angle, the value of whose sine is, similarly for other trigonometric functions (ii) The smallest numerical value, either positive or negative, of θ is called the principal value of the function

2 INVERSE TRIGONOMETRIC FUNCTIONS 9 (iii) Whenever no branch of an inverse trigonometric function is mentioned, we mean the principal value branch The value of the inverse trigonometic function which lies in the range of principal branch is its principal value Graph of an inverse trigonometric function The graph of an inverse trigonometric function can be obtained from the graph of original function by interchanging -ais and y-ais, ie, if (a, b) is a point on the graph of trigonometric function, then (b, a) becomes the corresponding point on the graph of its inverse trigonometric function It can be shown that the graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image (ie, reflection) along the line y Properties of inverse trigonometric functions sin (sin ) :, cos (cos ) : [0, ] ( ) :, cot (cot ) : ( 0, ) sec (sec ) : [0,] cosec (cosec ) :, {0} sin (sin ) : [,] cos (cos ) : [,] ( ) : R cot (cot ) : R sec (sec ) : R (,) cosec (cosec ) : R (,) sin cosec : R (,) cos sec : R (,)

3 0 MATHEMATICS cot : > 0 + cot : < 0 4 sin ( ) sin : [,] cos ( ) cos : [,] ( ) : R cot ( ) cot : R sec ( ) sec : R (,) cosec ( ) cosec : R (,) sin + cos : [,] + cot : R sec + cosec : R [,] 6 + y y y : y < y ; y y > + y 7 sin : cos Solved Eamples Short Answer (SA) : 0 : < < Eample Find the principal value of cos, for

4 INVERSE TRIGONOMETRIC FUNCTIONS Solution If cos θ, then cos θ Since we are considering principal branch, θ [0, ] Also, since > 0, θ being in the first quadrant, hence cos 6 Eample Evaluate sin Solution sin sin () 4 Eample Find the value of cos cos 6 Solution cos cos cos( ) 6 cos + 6 cos cos 6 6 Eample 4 Find the value of 9 8 Solution Eample Evaluate ( ( 4)) 8 8 Solution Since ( ), R, ( ( 4) 4 Eample 6 Evaluate: sec ( )

5 MATHEMATICS Solution sec ( ) [ sec ] + cos + Eample 7 Evaluate: sin cos sin Solution sin cos sin sin cos sin 6 Eample 8 Prove that (cot ) cot ( ) State with reason whether the equality is valid for all values of Solution Let cot θ Then cot θ or, θ θ So (cot ) θ cot θ cot cot cot( ) The equality is valid for all values of since and cot are true for R Eample 9 Find the value of sec y Solution Let y θ, where θ, So, θ y, which gives secθ 4 y Therefore, y 4 + y sec secθ Eample 0 Find value of (cos ) and hence evaluate Solution Let cos θ, then cos θ, where θ [0,] 8 cos 7

6 INVERSE TRIGONOMETRIC FUNCTIONS Therefore, (cos ) cos θ θ cosθ Hence cos Eample Find the value of sin cot Solution Let cot y Then cot y Now sin cot sin y siny cosy 0 69 Eample Evaluate Solution cos sin cos sin 4 sec 4 4 sec cos sin 4 since cot y< 0, so y, + cos 4 4 cos sin cos cos sin sin sin cos

7 4 MATHEMATICS Long Answer (LA) Eample Prove that sin 7 4 Solution Let sin θ, then sinθ, where θ, Thus θ 4, which gives θ 4 Therefore, sin 7 θ Eample 4 Prove that cot 7 + cot 8 + cot 8 cot Solution We have cot 7 + cot 8 + cot (since cot, if > 0) (since y 78 < )

8 INVERSE TRIGONOMETRIC FUNCTIONS (since y < ) 6 9 Eample Which is greater, or? cot Solution From Fig, we note that is an increasing function in the interval,, since > > This gives 4 4 Y > > > 4 > > () Eample 6 Find the value of / O /4 / X sin + cos( ) Solution Let and y so that and y Therefore, sin + cos( ) sin () + cos y y ( ) 7 + 6

9 6 MATHEMATICS Eample 7 Solve for, > 0 + Solution From given equation, we have Eample 8 Find the values of which satisfy the equation sin + sin ( ) cos Solution From the given equation, we have sin (sin + sin ( )) sin (cos ) sin (sin ) cos (sin ( )) + cos (sin ) sin (sin ( ) ) sin (cos ) ( ) + ( ) + ( ) 0 ( ) 0 0 or 0 or Eample 9 Solve the equation sin 6 + sin 6 Solution From the given equation, we have sin 6 sin 6

10 INVERSE TRIGONOMETRIC FUNCTIONS 7 sin (sin 6) sin sin 6 6 cos (sin 6 ) Squaring, we get 44 ± Note that is the only root of the equation as does not satisfy it Eample 0 Show that α β sin αcosβ 4 cosα+ sinβ Solution LHS α β 4 α β 4 since β α β + β α β + α β β α β +

11 8 MATHEMATICS α β β α β α α β α β + + α β + α β + + sin αcosβ cosα+ sinβ RHS Objective type questions Choose the correct answer from the given four options in each of the Eamples to 4 Eample Which of the following corresponds to the principal value branch of?,,, {0} (0, ) Solution is the correct answer Eample The principal value branch of sec is, {} 0 [ 0, ] (0, ),

12 INVERSE TRIGONOMETRIC FUNCTIONS 9 Solution is the correct answer Eample One branch of cos other than the principal value branch corresponds to, [,] (0, ) [, ] Solution is the correct answer Eample 4 The value of sin cos 4 is Solution is the correct answer 40 sin cos + sin cos 8 + sin cos sin sin sin sin 0 0 Eample The principal value of the epression cos [cos ( 680 )] is Solution is the correct answer cos (cos (680 )) cos [cos (70 40 )] cos [cos ( 40 )] cos [cos (40 )] 40 9 Eample 6 The value of cot (sin ) is + +

13 0 MATHEMATICS Solution is the correct answer Let sin θ, then sinθ cosec θ cosec θ + cot θ cotθ Eample 7 If 0 for some R, then the value of cot is 4 Solution is the correct answer We know + cot Therefore cot 0 cot 0 Eample 8 The domain of sin is [0, ] [, ], [, ] Solution is the correct answer Let sin θ so that sin θ Now sin θ, ie, which gives Eample 9 The principal value of sin is

14 INVERSE TRIGONOMETRIC FUNCTIONS 4 Solution is the correct answer sin sin sin sin sin Eample 0 The greatest and least values of (sin ) + (cos ) are respectively and 4 8 and and 4 4 and 0 4 Solution is the correct answer We have (sin ) + (cos ) (sin + cos ) sin cos sin sin 4 sin + ( sin ) 4 sin sin + 8 ( ) sin Thus, the least value is ie 6 8 and the Greatest value is + 4 6, ie 4 Eample Let θ sin (sin ( 600 ), then value of θ is

15 MATHEMATICS Solution is the correct answer 0 sin sin 600 sin sin 80 sin sin 4 sin sin sin sin sin sin Eample The domain of the function y sin ( ) is [0, ] (0, ) [, ] φ Solution is the correct answer y sin ( ) siny ie (since sin y ) 0 ie Eample The domain of y cos ( 4) is [, ] [0, ],,,, Solution is the correct answer y cos ( 4 ) cosy 4 ie 4 (since cos y ),, Eample 4 The domain of the function defined by f () sin + cos is

16 INVERSE TRIGONOMETRIC FUNCTIONS [, ] [, + ] (, ) φ Solution is the correct answer The domain of cos is R and the domain of sin is [, ] Therefore, the domain of cos + sin is R [,], ie, [, ] Eample The value of sin ( sin (6)) is sin Solution is the correct answer Let sin (6) θ, ie, sin θ 6 Now sin (θ) sinθ cosθ (6) (8) 96 Eample 6 If sin + sin y, then value of cos + cos y is 0 Solution is the correct answer Given that sin + sin y Therefore, cos + cos y cos + cos y Eample 7 The value of cos + 4 is Solution is the correct answer 4 4 cos + 4 +

17 4 MATHEMATICS Eample 8 The value of the epression sin [cot (cos ( ))] is 0 Solution is the correct answer sin [cot (cos 4 )] sin [cot ] sin sin Eample 9 The equation cot has no solution unique solution infinite number of solutions two solutions Solution is the correct answer We have cot 6 and + cot Adding them, we get ie, Eample 40 If α sin + cos β, then α, β α 0, β α, β α 0, β

18 INVERSE TRIGONOMETRIC FUNCTIONS Solution is the correct answer We have + sin sin + (sin + cos ) 0 sin + cos sin Eample 4 The value of (sec ) + cot (cosec ) is Solution is the correct answer (sec ) + cot (cosec ) sec (sec ) + cosec (cosec ) + EXERCISE Short Answer (SA) Find the value of cos cos Evaluate cos cos 6 Prove that cot cot Find the value of cot sin Find the value of 6 Show that ( ) + 4

19 6 MATHEMATICS 7 Find the real solutions of the equation ( ) + + sin Find the value of the epression sin + cos( ) 9 If (cos θ) ( cosec θ), then show that θ 4, where n is any integer 0 Show that cos sin 4 7 Solve the following equation cos( ) sin cot 4 Long Answer (LA) Prove that cos 4 Find the simplified form of 4 cos cos sin, where, Prove that Show that 8 77 sin sin sin sin cos 6 6 Prove that + sin Find the value of 4 9

20 INVERSE TRIGONOMETRIC FUNCTIONS 7 8 Show that 4 7 sin 4 and justify why the other value is ignored? 9 If a, a, a,,a n is an arithmetic progression with common difference d, then evaluate the following epression d d d d aa + aa + aa 4 + anan Objective Type Questions Choose the correct answers from the given four options in each of the Eercises from 0 to 7 (MCQ) 0 Which of the following is the principal value branch of cos?, (0, ) [0, ] (0, ) Which of the following is the principal value branch of cosec?, [0, ], If + cot, then equals, {0} 0 The value of sin cos is 7 0 0

21 8 MATHEMATICS 4 The domain of the function cos ( ) is [0, ] [, ] (, ) [0, ] The domain of the function defined by f () sin is [, ] [, ] [0, ] none of these 6 If cos sin + cos 0, then is equal to 0 7 The value of sin ( (7)) is equal to 7 96 sin 8 The value of cos cos is equal to 7 9 The value of the epression sec + sin is If + y 4, then cot + cot y equals 7 6 a a If sin a a the value of is cos 0 a, where a, ]0,, then a a a

22 INVERSE TRIGONOMETRIC FUNCTIONS 9 The value of cot 7 cos is The value of the epression cos is θ Hint : cosθ + cos θ 4 If, then + sin is equal to 4 0 If cos α + cos β + cos γ, then α (β + γ) + β (γ + α) + γ (α + β) equals The number of real solutions of the equation + cos cos (cos ) in, is 0 Infinite 7 If cos > sin, then < 0 < < > 0

23 40 MATHEMATICS Fill in the blanks in each of the Eercises 8 to 48 8 The principal value of cos is 9 The value of sin sin is 40 If cos ( + cot ) 0, then value of is 4 The set of values of sec is 4 The principal value of is 4 4 The value of cos cos is 44 The value of cos (sin + cos ), is 4 sin + cos The value of epression,when is 46 If y + sin for all, then < y < y 47 The result y + y is true when value of y is 48 The value of cot ( ) for all R in terms of cot is State True or False for the statement in each of the Eercises 49 to 49 All trigonometric functions have inverse over their respective domains 0 The value of the epression (cos ) is equal to sec The domain of trigonometric functions can be restricted to any one of their branch (not necessarily principal value) in order to obtain their inverse functions The least numerical value, either positive or negative of angle θ is called principal value of the inverse trigonometric function The graph of inverse trigonometric function can be obtained from the graph of their corresponding trigonometric function by interchanging and y aes

24 INVERSE TRIGONOMETRIC FUNCTIONS 4 n 4 The minimum value of n for which >, n N, is valid is 4 The principal value of sin cos sin is